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Semigroup Forum

, Volume 99, Issue 2, pp 260–292 | Cite as

\(\aleph _0\)-categoricity of semigroups

  • Victoria GouldEmail author
  • Thomas Quinn-Gregson
Open Access
Research Article
  • 178 Downloads

Abstract

In this paper we initiate the study of \(\aleph _0\)-categorical semigroups, where a countable semigroup S is \(\aleph _0\)-categorical if, for any natural number n, the action of its group of automorphisms \({\text {Aut}}(S)\) on \(S^n\) has only finitely many orbits. We show that \(\aleph _0\)-categoricity transfers to certain important substructures such as maximal subgroups and principal factors. We examine the relationship between \(\aleph _0\)-categoricity and a number of semigroup and monoid constructions, namely Brandt semigroups, direct sums, 0-direct unions, semidirect products and \({\mathcal {P}}\)-semigroups. As a corollary, we determine the \( \aleph _0\)-categoricity of an E-unitary inverse semigroup with finite semilattice of idempotents in terms of that of the maximal group homomorphic image.

Keywords

\(\aleph _0\)-categorical Semigroups Semidirect product 

1 Introduction

The concept of \(\aleph _0\)-categoricity is rooted in model theory. Let L be a first-order language and let T be a theory in L, that is, T is a set of sentences in L. The theory T is \(\aleph _0\)-categorical if T has exactly one countable model, up to isomorphism. In other words, there is a countable L-structure in which all the sentences of T are true, and is such that it is isomorphic to any other countable L-structure with the same property. We then say that an L-structure A is \(\aleph _0\)-categorical if \({\text {Th}}(A)\) is \(\aleph _0\)-categorical, where \({\text {Th}}(A)\) is the set of all sentences of L which are true in A. Thus, A is \(\aleph _0\)-categorical if it is determined up to isomorphism by its first-order properties. Morley’s celebrated categoricity theorem [25] was the impetus for the development of the rich area of model theory known as stability theory (see [28]).

It is a natural question to determine the \(\aleph _0\)-categorical members of any class of relational or algebraic structures: in our case, semigroups and monoids. This may be addressed without recourse to specialist model theory, in view of the following result, independently accredited to Engeler [8], Ryll-Nardzewski [37], and Svenonius [41], but commonly referred to as the Ryll-Nardzewski Theorem (RNT). Although stated in generality, we will apply it almost entirely in the context of semigroups, and semigroups with augmented structure (such as an identity, a partial order, or distinguished subsets).

Theorem 1.1

(Ryll-Nardzewski Theorem) A countable structure A is \(\aleph _0\)-categorical if and only if for each \(n\in {{\mathbb {N}}}\) the natural action of \({\text {Aut}}(A)\) on \(A^n\) has only finitely many orbits.

A number of authors have considered \(\aleph _0\)-categoricity for algebraic structures. Rosenstein [34] classified \(\aleph _0\)-categorical abelian groups, and an extensive overview of the results for groups is given in [1]. Baldwin and Rose [2] investigated \(\aleph _0\)-categoricity for rings. For semigroups per se, little is known in this context. This paper and a sequel [29] provide an introduction to the study of \(\aleph _0\)-categorical semigroups.

The notion of \(\aleph _0\)-categoricity has strong connections with that of homogeneity, where a structure is homogeneous if every isomorphism between finitely generated substructures extends to an automorphism. A homogeneous structure is \(\aleph _0\)-categorical if, for each \(n\in {\mathbb {N}}\), there is a finite bound on the number of non-isomorphic substructures which can be generated by n elements [18]. As we explain in Sect. 2 this leads to a wide class of \(\aleph _0\)-categorical semigroups arising from the second author’s work on the homogeneity of semigroups and, in particular, of semigroups of idempotents (bands) [30, 31, 32]. However, certainly not every \(\aleph _0\)-categorical semigroup is homogeneous, as shown in Sect. 2.

There are two main approaches to the study of \(\aleph _0\)-categoricity. The first half of this paper will be in line with the ‘preservation theorems’ approach. In particular, we shall investigate when the \(\aleph _0\)-categoricity of a semigroup passes to subsemigroups, quotients and certain direct sums. This is certainly a popular path to take: we mention here Grzegorczyk’s handy result that \(\aleph _0\)-categoricity (of a general structure) is preserved by finite direct products [16]. In [42], Waszkiewicz and Weglorz showed that \(\aleph _0\)-categoricity of a structure is preserved by Boolean extensions by \(\aleph _0\)-categorical Boolean algebras, a result later generalized by Schmerl [39] to filtered Boolean extensions. In [38] Sabbagh proved that the group \(\hbox {GL}_n(R)\) of invertible \(n\times n\) matrices over an \(\aleph _0\)-categorical ring R inherits \(\aleph _0\)-categoricity; the corresponding result for the semigroup \(M_n(R)\) of all \(n\times n\) matrices follows easily from Theorem 1.1. The final two sections fit into the ‘classification’ approach: determining the \(\aleph _0\)-categoricity of semigroups in certain classes built from \(\aleph _0\)-categorical components. In particular, we classify \(\aleph _0\)-categorical Brandt semigroups, 0-direct unions and certain \(\aleph _0\)-categorical semidirect products, including the case where the semigroup being acted upon is a finite semilattice.

A number of known classifications will be of use in our work, including the \(\aleph _0\)-categoricity of linear orders [33], which serve as examples of \(\aleph _0\)-categorical semilattices. For algebraic structures, the difficulty in achieving full classifications has long been apparent. Algebras in general do not display the high degree of global symmetry required for \(\aleph _0\)-categoricity, and knowledge of the group of automorphisms Aut(A) of an algebra A is not always important in determining A. However, significant results are available for groups and rings, largely because the underlying group structure forces a degree of symmetry. The former are of particular importance to this paper, since maximal subgroups of \(\aleph _0\)-categorical semigroups are \(\aleph _0\)-categorical (see Corollary 3.7). Our work on \(\aleph _0\)-categoricity for semigroups shows that even if we know that various constituent parts of a semigroup S, such as the maximal subgroups, are \(\aleph _0\)-categorical, it can be hard to determine when S itself is \(\aleph _0\)-categorical, due to the complex way in which the constituents are glued together.

Throughout the paper we develop tools for ascertaining \(\aleph _0\)-categoricity of semigroups, built on Theorem 1.1 with increasing degrees of complexity, which are made use of as follows. In Sect. 2 we show that any \(\aleph _0\)-categorical semigroup is periodic with bounded index and period (Corollary 2.3), and it can be cut into \(\aleph _0\)-categorical ‘slices’ which satisfy the additional property of being characteristically [0-]simple (Proposition 2.18). We also prove the existence of an \(\aleph _0\)-categorical nil semigroup that is not nilpotent, a situation that contrasts to that in ring theory [4] (Theorem 2.16). In Sect. 3 we develop a notion for subsemigroups of being relatively characteristic, which is somewhat weaker than the standard notion of being characteristic (i.e. preserved by all automorphisms). We are then able to demonstrate how \(\aleph _0\)-categoricity is inherited by maximal subgroups (Corollary 3.7), principal factors (Theorem 3.12), and certain other quotients (Corollary 3.11). We begin Sect. 4 by showing that a Brandt semigroup \({\mathcal {B}}^0[G;I]\) is \(\aleph _0\)-categorical if and only if G is an \(\aleph _0\)-categorical group (Theorem 4.2), a result which will be extended in the subsequent paper [29]. We also consider direct sums and 0-direct sums, determining when a direct sum of finite monoids or semigroups is \(\aleph _0\)-categorical (Theorem 4.4 and Proposition 4.5, respectively) and, via an analysis of 0-direct sums, when a primitive inverse semigroup is \(\aleph _0\)-categorical (Corollary 4.10). Finally in Sect. 5 we examine how \(\aleph _0\)-categoricity interacts with semidirect products, and with the construction of McAlister \({\mathcal {P}}\)-semigroups \({\mathcal {P}}={\mathcal {P}}(G,{\mathcal {X}},{{\mathcal {Y}}})\) in terms of the \(\aleph _0\)-categoricity of \(G,{{\mathcal {X}}}\) and \({{\mathcal {Y}}}\). In particular we show that if \({{\mathcal {Y}}}\) is finite then the \(\aleph _0\)-categoricity of \({\mathcal {P}}\) depends only on G (Theorem 5.12).

We denote the set of natural numbers (without 0) by \({{\mathbb {N}}}\) and we write \({{\mathbb {N}}}^0\) to mean \({{\mathbb {N}}}\cup \{ 0\}\). Given a semigroup S and \(n\in {\mathbb {N}}\), we let \(S^n\) denote both the set of n-tuples of S and all products of length n of elements from S; the meaning of the notation \(S^n\) should be clear from the context. Occasionally we make use of the convention that \(S^0=\emptyset \) denotes the set of 0-tuples of elements of S. The set of idempotents of a semigroup S will be denoted E(S). We take the convention that \(\emptyset \) is always an ideal of a semigroup. The identity bijection of a set X will be denoted by \(I_X\).

This article does not require any background in model theory, but we refer the reader to [18] for an introductory study, and to [9] for an overview of \(\aleph _0\)-categorical structures.

2 First examples of \(\aleph _0\)-categorical semigroups

As commented in the Introduction, our main tool in determining \(\aleph _0\)-categoricity is Theorem 1.1. An immediate consequence worth highlighting is:

Corollary 2.1

Finite semigroups are \(\aleph _0\)-categorical.

Recall from [18] that a structure M is uniformly locally finite (ULF) if there exists a function \(f:{\mathbb {N}}\rightarrow {\mathbb {N}}\) such that for every substructure T of M, if T has a generating set of cardinality at most n, then T has cardinality at most f(n). Rosenstein [34, Theorem 16] showed that an \(\aleph _0\)-categorical group is ULF, a result later proved for general structures:

Proposition 2.2

[18, Corollary 7.3.2] An \(\aleph _0\)-categorical structure is ULF.

We show below, by taking the example of an \(\omega \)-chain, that the converse to Proposition 2.2 need not hold. However, from the fact that an \(\aleph _0\)-categorical semigroup is ULF, we may immediately deduce the following:

Corollary 2.3

An \(\aleph _0\)-categorical semigroup S is periodic, with bounded index and period. Consequently, \(E(S)\ne \emptyset \) and \({{{\mathcal {D}}}}={{{\mathcal {J}}}}\).

A converse to Proposition 2.2 holds if we restrict our attention to homogeneous structures, where a structure M is homogeneous if every isomorphism between finitely generated substructures of M extends to an automorphism of M.

Proposition 2.4

[18, Corollary 7.4.2] A homogeneous ULF structure is \(\aleph _0\)-categorical.

Since McLean showed [24] that any band is ULF, we immediately have:

Corollary 2.5

Homogeneous bands and homogeneous semilattices are \(\aleph _0\)-categorical.

The homogeneity of both bands and inverse semigroups was studied by the second author, with results appearing in [31] (where a complete characterisation of homogeneous bands is given) and [32], respectively; see also the thesis [30] of the second author. Complete characterisations of homogeneous semilattices appear in [6, 7], and of \(\aleph _0\)-categorical linear orders in [33]. Let \({\mathcal {Q}}={{\mathbb {Q}}}\cup \{\infty \}\) be linearly ordered by extending the usual order in \({\mathbb {Q}}\) by adjoining a maximum element \(\infty \). It follows from these papers that \({\mathcal {Q}}\) is not a homogeneous semilattice but is an \(\aleph _0\)-categorical semilattice.

Further, an \(\omega \)-chain
$$\begin{aligned} x_1<x_2<\ldots \end{aligned}$$
or an inverse \(\omega \)-chain
$$\begin{aligned} x_1>x_2>\ldots \end{aligned}$$
are not \(\aleph _0\)-categorical. This contrasts starkly with the fact that an abelian group is \(\aleph _0\)-categorical if and only if it is periodic of bounded index [34].

At this stage it is convenient to fix some notation to help in the implementation of the RNT. Let AB be sets, \(n\in {{\mathbb {N}}}^0\), \(\phi :A\rightarrow B\) a map, \({\overline{a}}=(a_1,\dots ,a_n)\) an n-tuple of elements of A and \(M\subseteq A\). Then \({\overline{a}}\phi \) denotes the n-tuple of B given by \((a_1\phi ,\dots ,a_n\phi )\), and \(M\phi \) denotes the subset \(\{m\phi :m\in M\}\) of B.

Given a semigroup S and pair \({\overline{a}}=(a_1, \dots , a_n), \, {\overline{b}}=(b_1, \dots , b_n)\) of n-tuples of S, then we say that \({\overline{a}}\) is automorphically equivalent to/has the samen-automorphism type as\({\overline{b}}\) (in S) if there exists an automorphism \(\phi \) of S such that \({\overline{a}}\phi ={\overline{b}}\). We denote this relation on \(S^n\) by \({\overline{a}}\sim _{S,n} {\overline{b}}\), using the same notation for the restriction to subsets of \(S^n\). Hence, by the RNT, to prove that S is \(\aleph _0\)-categorical it suffices to show that, for each n, there exists a finite list of elements of \(S^n\) such that every element of \(S^n\) is automorphically-equivalent to an element of the list; equivalently, in any countably infinite list of elements of \(S^n\), we can find two distinct members that are automorphically-equivalent. We augment our notation as follows. Suppose that \({\overline{X}}\) is a finite tuple of elements of S. Let Aut\((S;{\overline{X}})\) denote the subgroup of Aut(S) consisting of those automorphisms which fix \({\overline{X}}\). We say that S is \(\aleph _0\)-categorical over\({\overline{X}}\) if Aut\((S;{\overline{X}})\) has only finitely many orbits in its action on \(S^n\) for each \(n\in {\mathbb {N}}\). We denote the resulting equivalence relation on \(S^n\) as \(\sim _{S,{\overline{X}},n}\), and a pair of \(\sim _{S,{\overline{X}},n}\)-equivalent n-tuples are said to be automorphically equivalent over\({\overline{X}}\).

Lemma 2.6 is a simple generalisation of [18, Exercise 7.3.1], and follows immediately from the RNT.

Lemma 2.6

Let S be a semigroup and \({\overline{X}}\) a finite tuple of elements of S. For any subset T of S, we have that \(|T^n/\sim _{S,n}|\) is finite for all \(n \in {\mathbb {N}}\) if and only if \(|T^n/\sim _{S,{\overline{X}},n}|\) is finite for all \(n\in {\mathbb {N}}\). In particular, S is \(\aleph _0\)-categorical if and only if S is \(\aleph _0\)-categorical over \({\overline{X}}\).

Example 2.7

Consider the countably infinite null semigroup N, with multiplication \(xy=0\) for all \(x,y\in N\). Since any permutation of the non-zero elements gives an isomorphism, it is clear that N is homogeneous. Clearly N is ULF, so that it is also \(\aleph _0\)-categorical. Indeed, N provides a good illustration of the RNT. A pair of n-tuples \({\overline{a}}=(a_1, \dots , a_n)\) and \( {\overline{b}}=(b_1, \dots , b_n)\) are automorphically equivalent if and only if the positions (if any exist) of the entries equal to 0 are the same in \({\overline{a}}\) and \( {\overline{b}}\), and for any \(1\le i,j\le n\) we have \(a_i=a_j\) if and only if \(b_i=b_j\). Since there are finitely many choices for each of these conditions, it follows that N is \(\aleph _0\)-categorical by the RNT.

It is worth formalising the points raised in the above argument, as they will be used throughout this paper. They are based on the following lemma, which may be proven by a simple counting argument.

Lemma 2.8

Let X be a set and \(\gamma _1,\dots ,\gamma _r\) be a finite list of equivalence relations on X with \(\gamma _1 \, \cap \, \gamma _2 \, \cap \cdots \cap \gamma _r\) contained in an equivalence relation \(\sigma \) on X. Then
$$\begin{aligned} |X/\sigma | \le \prod _{1\le i \le r} |X/\gamma _i|. \end{aligned}$$

We use the RNT in conjunction with Lemma 2.8 to prove that a semigroup S is \(\aleph _0\)-categorical in the following way.

Corollary 2.9

Let S be a semigroup and for each \(n\in {\mathbb {N}}\), let \(\gamma _1,\dots ,\gamma _{r(n)}\) be a finite list of equivalence relations on \(S^n\) such that \(S^n/\gamma _i\) is finite for each \(1\le i \le {r(n)}\) and
$$\begin{aligned} \gamma _1\cap \gamma _2 \cap \cdots \cap \gamma _{r(n)} \subseteq \, \sim _{S,n}. \end{aligned}$$
Then S is \(\aleph _0\)-categorical.

The equivalence relations \(\gamma _i\) appearing in Corollary 2.9 often arise from first considering a specific condition on a set (for example, the condition on \(({{\mathbb {N}}}^0)^n\) for some \(n\in {{\mathbb {N}}}\) that two n-tuples have any 0 entries in the same positions), and then by analysing the number of different ways in which the condition may be fulfilled (for example, considering the subsets of \(\{ 1,\ldots , n\}\) corresponding to 0 entries; see Example 2.10 (i) below). Where confusion is unlikely to arise, we may refer such equivalences in a less formal way, as follows. Suppose that we have an equivalence relation \(\sigma \) on \(S^n\) that arises from different ways in which a given condition may be fulfilled; if \(S^n/\sigma \) is finite, then we say the condition has finitely many choices.

Example 2.10

(i) If S is a semigroup with zero (as in Example 2.7), the equivalence \(\sim _0\) on \(S^n\) defined by the rule that
$$\begin{aligned} (a_1,\ldots , a_n)\sim _0 (b_1,\ldots , b_n)\Leftrightarrow \{ i:a_i=0\}=\{ i:b_i=0\} \end{aligned}$$
corresponds to the condition on n-tuples that they have the non-zero entries in the same positions, and this condition has \(2^n\) choices.
(ii) In a similar way to that of Example 2.7, given a set X, we may impose a condition on a pair of n-tuples of X which states that if a pair of entries in one of the tuples are equal then the same is true for the other tuple. Note that here we do not need to concern ourselves with 0 entries. Formally, we define an equivalence \(\natural _{X,n}\) on \(X^n\) by
$$\begin{aligned} (a_1,\dots ,a_n) \, \natural _{X,n} \, (b_1,\dots ,b_n) \text { if and only if } [a_i=a_j \Leftrightarrow b_i=b_j, \text { for each } i,j].\nonumber \\ \end{aligned}$$
(2.1)
It is clear that a pair of n-tuples \({\overline{a}}\) and \({\overline{b}}\) are \(\natural _{X,n}\)-equivalent if and only if there exists a bijection \(\phi :\{a_1,\dots ,a_n\} \rightarrow \{b_1,\dots ,b_n\}\) such that \(a_i\phi =b_i\). Moreover, the number of \(\natural _{X,n}\)-classes of \(X^n\) is equal to the number of ways of partitioning a set of size n, which is called the nth Bell number, denoted \(B_n\) (for a formulation, see [36]). In particular \(B_n\) is finite, for each \(n\in {\mathbb {N}}\). Note also that if S is a semigroup then
$$\begin{aligned} {\overline{a}} \, \sim _{S,n} \, {\overline{b}} \Rightarrow {\overline{a}} \, \natural _{S,n} \, {\overline{b}}. \end{aligned}$$

We will see in this paper that \(\aleph _0\)-categoricity ‘works well’ in conjunction with fixing finite sets of elements within semigroups. For example, in order to prove that S has finitely many n-automorphism types, it suffices to consider n-tuples of \(S {\setminus } T\), where T is finite:

Proposition 2.11

Let S be a semigroup and T a finite subset of S. Then S is \(\aleph _0\)-categorical if and only if \(|(S{\setminus } T)^n /\sim _{S,n}|\) is finite for each \(n\in {\mathbb {N}}\).

Proof

If S is \(\aleph _0\)-categorical then \(|S^n/\sim _{S,n}|\) is finite by the RNT, and thus so is \(|(S{\setminus } T)^n /\sim _{S,n}|\).

For the converse, we begin by fixing some notation. Let A be a subset of S and \({\overline{s}}=(s_1,\dots ,s_n)\) an n-tuple of S. Then we let
$$\begin{aligned} {\overline{s}}[A]:=\{k\in \{1,\dots ,n\}: s_k\in A\} \end{aligned}$$
be the set of positions of entries of \({\overline{s}}\) which lie in A. If \({\overline{s}}[A]=\{k_1,\dots ,k_r\}\) is such that \(k_1<k_2<\cdots <k_r\) then we obtain an r-tuple of A given by
$$\begin{aligned} {\overline{s}}^A:=(s_{k_1},\dots ,s_{k_r}). \end{aligned}$$
Let \(T=\{t_1,\dots ,t_r\}\) and take \({\overline{T}}=(t_1,\dots ,t_r)\in S^r\). Let \({\overline{a}}\) and \({\overline{b}}\) be n-tuples of S under the conditions that
  1. (1)

    \({\overline{a}}[T]={\overline{b}}[T]\) with \({\overline{a}}^T = {\overline{b}}^T\),

     
  2. (2)

    \({\overline{a}}^{S{\setminus } T}\) and \({\overline{b}}^{S{\setminus } T}\) are automorphically equivalent over \({\overline{T}}\).

     
Condition (1) has \((|T|+1)^n\) possibilities, which is finite since T is. Each \(|(S{\setminus } T)^m /\sim _{S,m}|\) is finite by our hypothesis, and so \(|(S{\setminus } T)^m/\sim _{S,{\overline{X}},m}|\) is also finite for each \(m\in {\mathbb {N}}\) by Lemma 2.6. Hence condition (2) has finitely many choices, and the total number of choices is therefore finite. By condition (2) there exists \(\phi \in \text {Aut}(S;{\overline{T}})\) with \({\overline{a}}^{S{\setminus } T}\phi ={\overline{b}}^{S{\setminus } T}\). Since \(t_i\phi =t_i\) for each \(1\le i \le r\) we have \({\overline{a}}^T\phi ={\overline{a}}^T={\overline{b}}^T\), and it follows that \({\overline{a}}\phi ={\overline{b}}\). The result is then immediate from Lemma 2.8. \(\square \)

For a semigroup S we let \(S^{\underline{1}}\) [\(S^0\)] denote S with an identity [zero] adjoined (whether or not S already has such an element). We note that, by convention, \(S^1\) denotes S with an identity adjoined if necessary. The next result follows from Proposition 2.11 and the fact that automorphisms of \(S^{\underline{1}}\) and \(S^0\) are exactly extensions of automorphisms of S.

Corollary 2.12

The following are equivalent for any semigroup S:
  1. (1)

    S is \(\aleph _0\)-categorical;

     
  2. (2)

    \(S^0\) is \(\aleph _0\)-categorical;

     
  3. (3)

    \(S^{\underline{1}}\) is \(\aleph _0\)-categorical.

     

In a similar fashion to that in Corollary 2.12 we can build new \(\aleph _0\)-categorical semigroups from given ingredients, provided the ingredients interact in a relatively simplistic way (see Proposition 2.20 below).

We now introduce an important notion for \(\aleph _0\)-categorical semigroups.

Definition 2.13

A subset [subsemigroup, ideal] A of a semigroup S is characteristic if it is invariant under automorphisms of S; that is, \(A\phi =A\) for all \(\phi \in \text {Aut}(S)\).

Clearly any subset A of a semigroup S is characteristic if and only if it is a union of \(\sim _{S,1}\)-classes, and if A is a characteristic subset of S then \(\langle A \rangle \) is a characteristic subsemigroup of S.

Let S be a semigroup with zero 0, and let \(n\in {{\mathbb {N}}}\). We say that S is nil of degree n if for all \(a\in S\) we have \(a^n=0\), and S is nilpotent of degree n if \(S^n=0\). If R is a ring, then we say R is nil/nilpotent (of degreen) if its underlying multiplicative semigroup is nil/nilpotent (of degree n).

Corollary 2.14

Let S be \(\aleph _0\)-categorical. Then
  1. (1)

    there are finitely many characteristic subsets of S;

     
  2. (2)

    any characteristic subsemigroup of S is \(\aleph _0\)-categorical;

     
  3. (3)

    any ideal \(S^m\) is characteristic and hence \(\aleph _0\)-categorical;

     
  4. (4)

    for some \(n\in {\mathbb {N}}\) we have \(S^n=S^{n+1}\), so that \(S^n=S^m\) for all \(m\ge n\);

     
  5. (5)

    with n as in (4), for any \(k<{\ell }\le n\) we have that \(S^{\ell }\) is an ideal of \(S^k\) and the Rees quotient \(S^k/S^{\ell }\) is \(\aleph _0\)-categorical;

     
  6. (6)

    with n as in (4), \(S/S^n\) is \(\aleph _0\)-categorical and nilpotent of degree n.

     

Proof

(1) follows from the fact that a subset is characteristic if and only if it is a union of \(\sim _{S,1}\) classes and (2) is immediate from the definition of characteristic subsemigroup. (3) is clear and then (4) is immediate from (1), (3), and the fact that \(S^n\supseteq S^{n+1}\).

For (5), observe that \(S^{\ell }\) is an ideal of \(S^k\). To see that \(S^k/S^{\ell }\) is \(\aleph _0\)-categorical, let \(m\in {{\mathbb {N}}}\) and consider a list of m-tuples of elements of \(S^k/S^{\ell }\). By Proposition 2.11 we may assume all of these elements are non-zero, and we may thus identify them with elements of \(S^k\). Since \(S^k\) is \(\aleph _0\)-categorical we may find a distinct pair \((a_1,\ldots , a_m)\) and \((b_1,\ldots , b_m)\) in our list and \(\phi \in {\text {Aut}}(S^k)\) such that \(a_i\phi =b_i\) for \(1\le i\le m\). It is easy to see that \(\phi \) induces an automorphism \(\phi '\) of \(S^k/S^{\ell }\), and regarded as m-tuples of \(S^k/S^{\ell }\), we have \((a_1,\ldots , a_m)\phi '=(b_1,\ldots , b_m)\). Hence by Proposition 2.11, \(S^k/S^{\ell }\) is \(\aleph _0\)-categorical.

For (6), observe \(S/S^n\) is nilpotent of degree n. The rest of the statement follows from (5). \(\square \)

Corollary 2.14 shows that any \(\aleph _0\)-categorical semigroup is associated with a nilpotent one. A major result for \(\aleph _0\)-categorical rings states that any \(\aleph _0\)-categorical nil ring of degree n is nilpotent of degree n [4]. We show that the corresponding result is not true for semigroups, by constructing a countably infinite \(\aleph _0\)-categorical commutative semigroup S such that S is nil of degree 2 and \(S=S^2\), so that certainly S is non-nilpotent.

A commutative semigroup S, nil of degree 2 such that \(S=S^2\) is called a zs-semigroup. Some progress has been made in understanding the structure of zs-semigroups, including [20] and [12]. In [12], a simple example of a zs-semigroup is constructed, which is very similar to that given below. However, we need to start with a countable atomless Boolean algebra in order to ensure the resulting zs-semigroup is \(\aleph _0\)-categorical.

Remark 2.15

There exists a unique, up to isomorphism, countable atomless Boolean algebra B [13, Theorem 10].1 Since atomless Boolean algebras are axiomatisable, it follows that B is \(\aleph _0\)-categorical. We can construct B in a number of ways, including the Lindenbaum algebra of propositional logic [21, Chapter 6]. For our purposes it is convenient to use the construction given in [15, Corollary 23] via certain subsets of [0, 1]. Let B be the set of all subsets of \([0,1]\cap {\mathbb {Q}}\) of the form
$$\begin{aligned} (a_0]\cup (b_1,a_1]\cup \cdots \cup (b_{n-1},a_{n-1}] \end{aligned}$$
where \(a_0, b_1,a_1,\ldots , a_{n-1}\in {\mathbb {Q}}\) with \( 0<a_0<b_1<a_1<\cdots<b_{n-1}<a_{n-1}\le 1\) and \((a_0]\) may be missing. It is clear that B forms a subalgebra of the Boolean algebra of subsets of \((0,1]\cap {\mathbb {Q}}\) and is atomless. Moreover, if \(A = (a_0]\cup (b_1,a_1]\cup \cdots \cup (b_{n-1},a_{n-1}]\) is a non-empty element of B then, taking any \(x\in (b_1,a_1)\cap {\mathbb {Q}}\), we have that
$$\begin{aligned} A=\big ((a_0]\cup (b_1,x]\big ) \cup \big ((x,a_1]\cup \cdots \cup (b_{n-1},a_{n-1}]\big ) \end{aligned}$$
and
$$\begin{aligned} \big ((a_0]\cup (b_1,x]\big ) \cap \big ((x,a_1]\cup \cdots \cup (b_{n-1},a_{n-1}]\big )=\emptyset . \end{aligned}$$

Theorem 2.16

There is an \(\aleph _0\)-categorical non-nilpotent, nil semigroup.

Proof

We show there is an \(\aleph _0\)-categorical zs-semigroup that is not nilpotent.

Let \(B=(B,\wedge , \vee , 0,1)\) be the countable atomless Boolean algebra and let \(B^*=B{\setminus } \{0\}\). Define an operation \(+\) on \(B^*\) by
$$\begin{aligned} A+A' = \left\{ \begin{array}{ll} A \vee A' &{}\quad \text {if } A\wedge A'=0,\\ 1 &{}\quad \text {else. } \end{array} \right. \end{aligned}$$
Since \(\vee \) is an associative operation and \(\wedge \) distributes over \(\vee \), we have for any \(A_1,A_2,A_3\in B^*\),
$$\begin{aligned} (A_1+A_2)+A_3&= \left\{ \begin{array}{ll} (A_1\vee A_2)+A_3 &{}\quad \text {if } A_1\wedge A_2=0,\\ 1+A_3 &{}\quad \text {else, } \end{array} \right. \\&= \left\{ \begin{array}{ll} (A_1\vee A_2)\vee A_3 &{}\quad \text {if } A_1\wedge A_2=0 \text { and } (A_1 \vee A_2)\wedge A_3=0,\\ 1 &{}\quad \text {else, } \end{array} \right. \\&= \left\{ \begin{array}{ll} A_1\vee (A_2\vee A_3) &{}\quad \text {if } A_1\wedge A_2=0 \text { and } (A_1 \wedge A_3) \vee (A_2\wedge A_3)=0,\\ 1 &{}\quad \text {else, } \end{array} \right. \\&= \left\{ \begin{array}{ll} A_1\vee (A_2\vee A_3) &{}\quad \text {if } A_1\wedge A_2=0, A_1 \wedge A_3=0, \text { and } A_2\wedge A_3=0,\\ 1 &{}\quad \text {else, } \end{array} \right. \\&= \left\{ \begin{array}{ll} A_1\vee (A_2\vee A_3) &{}\quad \text {if } A_2\wedge A_3=0 \text { and } A_1 \wedge (A_2 \vee A_3)=0,\\ 1 &{}\quad \text {else,} \end{array} \right. \\&= \left\{ \begin{array}{ll} A_1+ (A_2\vee A_3) &{}\quad \text {if } A_2\wedge A_3=0,\\ A_1+ 1 &{}\quad \text {else, } \end{array} \right. \\&= A_1+(A_2+A_3). \end{aligned}$$
Hence \((B^*,+)\) forms a semigroup, and is commutative by the commutativity of \(\wedge \) and \(\vee \). Note that \(A+1 = A\vee 1 = 1\) for all \(A\in B^*\), so that 1 is the zero of \((B^*,+)\). If \(A\in B^*\), then \(A+A=1\) as \(A\wedge A=A\), and so \((B^*,+)\) forms a nil semigroup of degree 2. Moreover, by the last part of Remark 2.15 for any \(A\in B^*\) we have \(A=A_1 \vee A_2\) for some \(A_1,A_2\in B^*\) such that \(A_1 \wedge A_2=0\). Hence \(A=A_1+A_2\), and so \((B^*,+)\) forms a zs-semigroup. Let us denote this semigroup by [B]. It is clear that if \(\theta \) is an automorphism of B, then \(\theta |_{B^*}\) is an automorphism of \(B^*\). Since B is \(\aleph _0\)-categorical, so is [B]. \(\square \)

We can say a little more: it is easy to build an example of an \(\aleph _0\)-categorical commutative nil semigroup that is nil of degree 2 and nilpotent but not nilpotent of degree 2.

Example 2.17

Let A be a countably infinite set, and let u, 0 be distinct symbols not in A. Let \(C=A\cup \{ 0,u\}\) and define a binary operation on C by letting the only non-zero products be \(ab=u\) where \(a,b\in A\) and \(a\ne b\). It is easy to see that C is a commutative semigroup, nil of degree 2 and nilpotent of degree 3. That C is \(\aleph _0\)-categorical follows easily from Proposition 2.11.

We say that a semigroup S is characteristically simple if it has no characteristic ideals other than \(\emptyset \) or itself. Similarly, we say that a semigroup S with 0 is characteristically 0-simple if it has no characteristic ideals, other than \(\emptyset \), \(\{ 0\}\) and itself.

Proposition 2.18

Let S be an \(\aleph _0\)-categorical semigroup. Then S is the union of a finite chain of characteristic subsemigroups
$$\begin{aligned} S=S_0\supset S_1\supset \ldots \supset S_n \end{aligned}$$
such that for \(0\le i\le n-1\), \(S_{i-1}\) is a characteristic ideal of \(S_i\) and the Rees quotients \(S_i/S_{i+1}\) are \(\aleph _0\)-categorical and characteristically 0-simple, and \(S_n\) is characteristically simple.

Proof

For an \(\aleph _0\)-categorical semigroup S, let \(\tau (S)\) denote \(|S/\sim _{S,1}|\). Let U be a characteristic subsemigroup of S. Notice that for any \(n\in {{\mathbb {N}}}\) and \({\overline{u}},{\overline{v}}\in U^n\), if \({\overline{u}}\sim _{S,n} {\overline{v}}\), then \({\overline{u}}\sim _{U,n} {\overline{v}}\), since \(U\phi =U\) for any \(\phi \in {\text {Aut}}(S)\). Hence if \(U\ne S\) is a characteristic subsemigroup of S then U is \(\aleph _0\)-categorical by Corollary 2.14 (2) and \(\tau (U)<\tau (S)\).

We proceed by induction on \(\tau (S)\). If \(\tau (S)=1\), then certainly there are no proper characteristic ideals of S, so that the result is true with \(n=0\).

Suppose now that for any \(\aleph _0\)-categorical semigroup T with \(\tau (T)<\tau (S)\) the result holds. Let T be a maximal proper characteristic ideal of S. If \(T=\emptyset \) then we are done. Suppose therefore that \(T\ne \emptyset \); the proof that S / T is \(\aleph _0\)-categorical follows as in (5) of Corollary 2.14 (see also (1) of Corollary 3.11).

If U is a proper characteristic ideal of S / T, then either \(U=\{ 0\}\), or \(U{\setminus } \{ 0\}\cup T\) is an ideal of S. Since \(U{\setminus } \{0\}\) is a union of \(\sim _{S/T,1}\)-classes and hence of \(\sim _{S,1}\)-classes, we have that \((U{\setminus }\{ 0\})\cup T\) is a characteristic ideal of S strictly containing T, a contradiction. Thus S / T is characteristically 0-simple.

From the first part of the proof we have \(\tau (T)<\tau (S)\), so that, applying the result for T, we deduce the required sequence of ideals for S. \(\square \)

The following example is clear.

Example 2.19

Let S be a semigroup. Then E(S) and \({\text {Reg}}(S)\) form characteristic subsets of S, where Reg(S) is the set of regular elements of S. The following subsemigroups of S (where they exist) are characteristic:
$$\begin{aligned} \langle E(S)\rangle , \{ 1\}, \{ 0\}, \langle {\text {Reg}}(S)\rangle . \end{aligned}$$
If S is commutative, then \(E(S) = \langle E(S) \rangle \) forms a band and Reg\((S) = \langle {\text {Reg}}(S) \rangle \) forms a semilattice of abelian groups, that is, a commutative Clifford semigroup (see [19, Chapter IV]).

We address the \(\aleph _0\)-categoricity of Clifford semigroups in the sequel [29]. Proposition 2.20 below gives a taster of the results for Clifford semigroups, in the special case where the connecting homomorphisms are trivial.

Proposition 2.20

Let \(S=\bigcup _{i\in Y}S_i\) be a finite chain of semigroups such that for any \(i>j, s_i\in S_i, s_j\in S_j\) we have \(s_is_j=s_j=s_js_i\). If each \(S_i\) is \(\aleph _0\)-categorical then S is \(\aleph _0\)-categorical. Moreover, if each \(S_i\) is characteristic (for example, if it is a non-trivial group) then the converse holds.

Proof

Let us refer to the \(S_i\) (\(i\in Y\)) as the components of S.

Suppose each \(S_i\) is \(\aleph _0\)-categorical. Let \(n\in {{\mathbb {N}}}\) and notice that in any infinite list of elements of \(S^n\) we can pick a sublist \((a_1^i,\ldots , a_n^i)\) such that for any \(1\le \ell \le n\) the elements \(a_{\ell }^1,a_{\ell }^2,\ldots \) all lie in the same component of S. Without loss of generality, suppose that \(a_1^k,\ldots , a^k_{j_1}\in S_{i_1}, a^k_{j_1+1},\ldots , a^k_{j_2}\in S_{i_2},\ldots , a^k_{j_{u-1}+1},\ldots , a^k_{n}\in S_{i_u}\). Since each \(S_k\) is \(\aleph _0\)-categorical, we may find an \(i<j\) and \(\phi _{\ell }\in {\text {Aut}}(S_{i_{\ell }}), 1\le \ell \le u\), such that \(\phi '=\bigcup _{1\le \ell \le u}\phi _{\ell }\) takes \((a^i_1,\ldots , a^i_n)\) to \((a^j_1,\ldots , a^j_n)\). For any \(t\in Y{\setminus } \{ i_1,\ldots , i_u\}\), let \(\phi _t=I_{S_t}\). It is easy to see that \(\phi =\bigcup _{i\in Y}\phi _i\) lies in \({\text {Aut}}(S)\) and clearly takes \((a^i_1,\ldots , a^i_n)\) to \((a^j_1,\ldots , a^j_n)\).

The converse is clear. \(\square \)

Finally in this section we make a comment concerning chains of [one-sided] ideals of S. Recall that in a partially ordered set L, an element ucovers an element v, written \(v\prec u\), if \(v<u\) and for all w with \(v\le w\le u\) we have \(v=w\) or \(w=u\). If we have a chain of elements in L such that each element covers its predecessor, then we call this a covering chain.

Proposition 2.21

Let S be an \(\aleph _0\)-categorical semigroup in which the principal right [left, two-sided] ideals form a chain. Then there are no infinite ascending or descending covering chains of principal right [left, two-sided] ideals.

Proof

We argue for ascending chains of principal right ideals, the other cases being similar.

Suppose that \(a_1,a_2,\ldots \in S\) and
$$\begin{aligned} a_1S^1\prec a_2S^1\prec \ldots . \end{aligned}$$
Then \((a_1,a_i)\) for \(i\in {{\mathbb {N}}}\) lie in different 2-automorphism types, a contradiction. \(\square \)

As the case of a dense linear order shows, we cannot expect to have full ascending or descending chain conditions on ideals in \(\aleph _0\)-categorical semigroups.

3 Inherited categoricity

We remarked in Corollary 2.14 that \(\aleph _0\)-categoricity is inherited by characteristic subsemigroups. We note that \(\aleph _0\)-categoricity is not inherited by every subsemigroup, and an example for groups is given by Olin in [27]. However, the condition that a subsemigroup be characteristic to inherit \(\aleph _0\)-categoricity is too restrictive; since many key subsemigroups, such as maximal subgroups and principal ideals, are not necessarily characteristic. The components in a finite chain of groups as in Proposition 2.20 are, but this relies on the chain being finite. We thus study a weaker condition for a subsemigroup that still guarantees the preservation of \(\aleph _0\)-categoricity.

Definition 3.1

Let S be a semigroup and, for some fixed \(t\in {\mathbb {N}}\), let \(\{{\overline{X}}_i:i\in I\}\) be a collection of t-tuples of S. Let \(\{A_i:i\in I\}\) be a collection of subsets [subsemigroups, ideals] of S with the property that for any automorphism \(\phi \) of S such that there exists \(i,j\in I\) with \({\overline{X}}_i \phi = {\overline{X}}_j\), then \(\phi |_{A_i}\) is a bijection from \(A_i\) onto \(A_j\). Then we call \({\mathcal {A}}=\{(A_i,{\overline{X}}_i):i\in I\}\) a system oft-pivoted pairwise relatively characteristic (t-pivoted p.r.c.) subsets [subsemigroups, ideals] of S. The t-tuple \({\overline{X}}_i\) is called the pivot of \(A_i\) (\(i\in I\)). If \(|I|=1\) then, letting \(A_1=A\) and \({\overline{X}}_1={\overline{X}}\), we write \(\{(A,{\overline{X}})\}\) simply as \((A,{\overline{X}})\), and call A an \({\overline{X}}\)-pivoted relatively characteristic (\({\overline{X}}\)-pivoted r.c.) subset [subsemigroup, ideal] of S.

Clearly if \(\{(A_i,{\overline{X}}_i):i\in I\}\) forms a system of t-pivoted p.r.c. subsets of S and J is a subset of I then \(\{(A_j,{\overline{X}}_j):j\in J\}\) is also a system of t-pivoted p.r.c. subsets of S. In particular, each \(A_i\) is an \({\overline{X}}_i\)-pivoted r.c. subset of S. Moreover, if A is an \({\overline{X}}\)-pivoted r.c. subset of S then A is a union of orbits of the set of automorphisms of S which fix \({\overline{X}}\), since if \(a\in A\) and \(\phi \in \text {Aut}(S)\) fixes \({\overline{X}}\) then \(A\phi =A\), so that \(a\phi \in A\).

Definition 3.1 has strong links with the model theoretic concept of definability, and we refer the reader to the introduction of [10] for a background into these links. In fact much of the work in this section could be given in terms of definable sets, but in keeping with our algebraic viewpoint it is more natural to use Definition 3.1.

Lemma 3.2

Let S be a semigroup and, for some fixed \(t\in {\mathbb {N}}\), let \(\{{\overline{X}}_i:i\in I\}\) be a collection of t-tuples of S. Then for any collection \(\{A_i:i\in I\}\) of subsets of S, the following are equivalent:
  1. (1)

    \(\{(A_i,{\overline{X}}_i):i\in I\}\) is a system of t-pivoted p.r.c. subsets [subsemigroups, ideals] of S;

     
  2. (2)

    if \(\phi \in {\text {Aut}}(S)\) is such that there exists \(i,j\in I\) with \({\overline{X}}_i\phi = {\overline{X}}_j\), then \(A_i\phi \subseteq A_j\).

     

Proof

This follows immediately from applying the definitions, and the fact that if \(\phi \in {\text {Aut}}(S)\) and \({\overline{X}}_i\phi = {\overline{X}}_j\), then \(\phi ^{-1}\in {\text {Aut}}(S)\) and \({\overline{X}}_j\phi ^{-1} = {\overline{X}}_i\). \(\square \)

Consequently, if \(\{(A_i,{\overline{X}}_i):i\in I\}\) is a system of t-pivoted p.r.c. subsets of a semigroup S then \(\{(\langle A_i \rangle ,{\overline{X}}_i):i\in I\}\) forms a system of t-pivoted p.r.c. subsemigroups of S. For if \(\phi \in \text {Aut}(S)\) is such that \({\overline{X}}_i\phi ={\overline{X}}_j\) for some \(i,j\in I\) then \(A_i\phi =A_j\), and so \(\langle A_i \rangle \phi \subseteq \langle A_j \rangle \). The result follows by Lemma 3.2.

Notation

Given a pair of tuples \({\overline{a}}=(a_1,\dots ,a_n)\) and \({\overline{b}}=(b_1,\dots ,b_m)\), we denote \(({\overline{a}},{\overline{b}})\) as the \((n+m)\)-tuple given by
$$\begin{aligned} (a_1,\dots ,a_n,b_1,\dots ,b_m). \end{aligned}$$

Proposition 3.3

Let S be an \(\aleph _0\)-categorical semigroup and \(\{(A_i,{\overline{X}}_i):i\in I\}\) be a system of t-pivoted p.r.c. subsets of S. Then \(\{|A_i| : i \in I\}\) is finite. If, further, each \(A_i\) forms a subsemigroup of S, then \(\{A_i: i \in I\}\) is finite, up to isomorphism, with each \(A_i\) being \(\aleph _0\)-categorical.

Proof

Suppose for some \(i\ne j\) we have \({\overline{X}}_i \, \sim _{S,t} \, {\overline{X}}_j\) via \(\phi \in \text {Aut}(S)\), say. Then \(A_i\phi =A_j\) and it follows that both \(|\{|A_i|:i\in I\}|\) and number of non-isomorphic elements of \(\{A_i:i\in I\}\) is bound by the number of t-automorphism types of S, which is finite by the \(\aleph _0\)-categoricity of S.

Suppose \(A_i\) forms a subsemigroup of S. Let \({\overline{X}}_i=(x_{i1},\dots ,x_{it})\), and suppose \({\overline{a}}=(a_1,\dots ,a_n)\) and \( {\overline{b}}=(b_1,\dots ,b_n)\) are a pair of n-tuples of \(A_i\) such that \(({\overline{a}},{\overline{X}}_i) \, \sim _{S,n+t} \, ({\overline{b}},{\overline{X}}_i)\) via \(\phi \in \text {Aut}(S)\), say. Then \({\overline{X}}_i\phi ={\overline{X}}_i\) and so \(\phi |_{ A_i }\) is an automorphism of \(A_i\) as \((A_i,{\overline{X}}_i)\) is a t-pivoted r.c. subsemigroup. Moreover, \({\overline{a}}\phi |_{A_i} ={\overline{a}}\phi = {\overline{b}}\) and so \({\overline{a}} \, \sim _{A_i,n} \, {\overline{b}}\). We have thus shown that
$$\begin{aligned} | A_i^n / \sim _{A_i,n}| \le |S^{n+t}/\sim _{S,n+t}|< \aleph _0 \end{aligned}$$
for each \(n\in {\mathbb {N}}\), since S is \(\aleph _0\)-categorical. Hence \(A_i\) is \(\aleph _0\)-categorical by the RNT. \(\square \)

Corollary 3.4

Let S be an \(\aleph _0\)-categorical semigroup. Then there are only finitely many principal [left, right] ideals, up to isomorphism, and these are all \(\aleph _0\)-categorical.

Proof

We show that \(\{(S^1aS^1,a):a\in S\}\) forms a system of 1-pivoted p.r.c. subsets of S. To see this, let \(\phi \in \text {Aut(S)}\) be such that \(a\phi =b\), and let \(x\in S^1aS^1\). Then there exists \(u,v\in S^1\) with \(x=uav\), and so by interpreting \(1\phi \) as 1 we have
$$\begin{aligned} x\phi =(u\phi )(a\phi )(v\phi )=(u\phi )b(v\phi )\in S^1bS^1, \end{aligned}$$
and the result follows by Lemma 3.2. A similar result holds for principal left/right ideals. \(\square \)
Motivated by Green’s relations, we now give a method for constructing systems of t-pivoted p.r.c. subsets of a semigroup via certain equivalence relations. Let \(\phi :S\rightarrow T\) be an isomorphism between semigroups S and T, and \(\tau _S\) and \(\tau _T\) be equivalence relations on S and T, respectively. We call \(\tau _S\) and \(\tau _T\)preserved by\(\phi \) if \(a \, \tau _S \, b\) if and only if \(a\phi \, \tau _T \, b\phi \) for each \(a,b\in S\). This is clearly equivalent to
$$\begin{aligned} (x{\tau _S})\phi =(x\phi ){\tau _T} \quad (\forall x\in S), \end{aligned}$$
where \(\phi \) is applied pointwise, and yet again to \({\phi }': S/\tau _S\rightarrow T/\tau _T\) given by
$$\begin{aligned}{}[x]_{\tau _S}{\phi }'=[x\phi ]_{\tau _T} \quad (\forall x\in S) \end{aligned}$$
being a well-defined bijection. If \(S=T\), so that \(\phi \) is an automorphism of S, then we say that \(\tau _S\) is preserved by\(\phi \).
Note that if \(\tau \) is an equivalence relation on a semigroup S then
$$\begin{aligned} \text {Aut}(S)[\tau ]:=\{\phi \in \text {Aut}(S):\tau \text { is preserved by } \phi \} \end{aligned}$$
is a subgroup of Aut(S).

If \(\text {Aut}(S)=\text {Aut}(S)[\tau ]\) then we call \(\tau \)preserved by automorphisms (of S).

Example 3.5

  1. (1)

    For any semigroup S, if U is a characteristic subset, then \(\langle U\times U\rangle \) is preserved by automorphisms.

     
  2. (2)
    If S is an inverse semigroup, then the least group congruence \(\sigma \) on S given by
    $$\begin{aligned} a \, \sigma \, b \Leftrightarrow (\exists e\in E(S)) \quad ea=eb, \end{aligned}$$
    is preserved by automorphisms.
     
  3. (3)

    If \(\rho \) is a relation preserved by automorphisms, then so too are the congruences \(\rho ^{\sharp }=\langle \rho \rangle \) and \(\rho ^{\flat }\), where \(\rho ^{\flat }\) is the largest congruence contained in \(\rho \).

     
  4. (4)

    If S is an inverse semigroup then \(\mu \), the maximum idempotent-separating congruence on S, is preserved by automorphisms.

     

Proof

Statements (1) and (3) are clear. We remark that if \(\sigma \) is given by the formula in (2), then \(\sigma =\langle E(S)\times E(S)\rangle \), and for (4) we require the fact that for any inverse semigroup S, we have \(\mu ={\mathcal {H}}^{\flat }\) [19, Proposition 5.3.7]. \(\square \)

The following lemma is then immediate from Proposition 3.3.

Lemma 3.6

Let S be a semigroup and \(\tau \) be an equivalence relation on S, preserved by automorphisms of S. Then \(\{(x\tau ,x):x\in S\}\) forms a system of 1-pivoted p.r.c. subsets of S. Hence if S is \(\aleph _0\)-categorical then there are only finitely many cardinalities of \(\tau \)-classes and only finitely many subsemigroup \(\tau \)-classes, up to isomorphism.

Corollary 3.7

Let S be an \(\aleph _0\)-categorical semigroup. Then for any of Green’s relations \({\mathcal {K}}\), we have
$$\begin{aligned} |\{ |K_a|:a\in S\}|<\aleph _0. \end{aligned}$$
Moreover, if U is a transversal of the set of \({\mathcal {K}}\)-classes that are subsemigroups, there are only finitely many \(K_u\)-classes \((u\in U)\), up to isomorphism, and each \(K_u\) is \(\aleph _0\)-categorical.

In particular, there are only finitely many maximal subgroups, up to isomorphism, and each of these is \(\aleph _0\)-categorical.

Proof

Each Green’s relation is preserved by automorphisms. Consequently, for any semigroup S and any \(K\in \{ {\mathcal {R}},{{{\mathcal {L}}}},{{{\mathcal {H}}}},{{{\mathcal {D}}}},{{{\mathcal {J}}}}\}\), we have \(\{(K_a,a):a\in S\}\) as a system of 1-pivoted p.r.c. subsets of S. The result then follows from Lemma 3.6. \(\square \)

A similar statement to the above also holds for Green’s *-relations, and Green’s \(\widetilde{\phantom {{\mathcal {R}}}}\)-relations [14]. It is worth exercising some caution here. In the corollary above the maximal subgroups are \(\aleph _0\)-categorical semigroups, while earlier investigations into the \(\aleph _0\)-categoricity of groups considered them as a set with a single binary operation, a single unary operation (inverse), and a single constant (the identity). However, since a semigroup automorphism of a group is necessarily a group automorphism, it follows from the RNT that our two concepts of \(\aleph _0\)-categoricity of a group coincide, and we can write \(\aleph _0\)-categorical group without ambiguity.

Much like the situation with characteristic subsets, for results relating to inherited \(\aleph _0\)-categoricity of quotients we require only that congruences are preserved by all automorphisms fixing a finite number of elements. This leads us to the following definition.

Definition 3.8

Let \(\tau \) be an equivalence relation on a semigroup S and \({\overline{X}}\) be an n-tuple of S for some \(n\in {\mathbb {N}}^0\). We say that \(\tau \) is \({\overline{X}}\)-relatively automorphism preserved (\({\overline{X}}\)-r.a.p.) with pivot\({\overline{X}}\), if whenever \(\phi \in \text {Aut}(S)\) is such that \({\overline{X}}\phi ={\overline{X}}\), then \(\phi \) preserves \(\tau \).

If \({\overline{X}}\) is the empty tuple then \(\tau \) being \({\overline{X}}\)-r.a.p. is equivalent to \(\tau \) being preserved by automorphisms. Moreover, as with \({\overline{X}}\)-pivoted r.c. subsets, there exists connections between definable sets of ordered pairs of a semigroup and \({\overline{X}}\)-r.a.p. equivalence relations.

Lemma 3.9

Let S be a semigroup, let \({\overline{X}}\in S^t\) for some \(t\in {{\mathbb {N}}}^0\), and \(\tau \) an \({\overline{X}}\)-r.a.p. equivalence relation on S. For each \(a\in S\), let \({\overline{X}}_a\) be the \((t+1)\)-tuple given by \(({\overline{X}},a)\). Then \(\{(a\tau ,{\overline{X}}_a):a\in S\}\) forms a system of \((t+1)\)-pivoted p.r.c. subsets of S.

Proof

Let \(\phi \) be an automorphism of S such that \({\overline{X}}_a\phi ={\overline{X}}_b\) for some \(a,b\in S\). Then \({\overline{X}}\phi =X\) so that \(\tau \) is preserved by \(\phi \), and \(a\phi =b\). Hence
$$\begin{aligned} (a{\tau })\phi =b\tau . \end{aligned}$$
\(\square \)

Note that by letting \({\overline{X}}\) be the empty tuple in the lemma above we recover the first statement in Lemma 3.6.

Our next aim is to use the results above to assess when the \(\aleph _0\)-categoricity of a semigroup passes to its quotients.

Proposition 3.10

Let S be an \(\aleph _0\)-categorical semigroup, let \({\overline{X}}\in S^t\) for some \(t\in {{\mathbb {N}}}^0\), and \(\rho \) an \({\overline{X}}\)-r.a.p. congruence on S. Then \(S/\rho \) is \(\aleph _0\)-categorical.

Proof

Suppose \({\overline{X}}\in S^t\) and let \({\overline{a}}=(a_1 \rho , \dots , a_n \rho )\) and \({\overline{b}}=(b_1 \rho , \dots , b_n \rho )\) be a pair of n-tuples of \(S/\rho \) such that \((a_1,\dots ,a_n,{\overline{X}}) \sim _{S,n+t} (b_1,\dots ,b_n,{\overline{X}})\) via \(\phi \in \text {Aut}(S)\), say. Then \({\overline{X}}\phi ={\overline{X}}\), so that \(\rho \) is preserved by the automorphism \(\phi \), and there is thus an automorphism \(\psi \) of \(S/\rho \) given by
$$\begin{aligned}{}[a]_{\rho } \psi = [a\phi ]_{ \rho } \quad (a\rho \in S/\rho ). \end{aligned}$$
Since \([a_k]_{ \rho } \psi = [a_k \phi ]_{\rho }=[b_k ]_{\rho }\) for each \(1\le k \le n\), we have \({\overline{a}} \sim _{S/\rho , n} {\overline{b}}\), and so
$$\begin{aligned} |(S/ \rho )^n / \sim _{S/ \rho , n}| \le |S^{n+t} / \sim _{S,n+t}|<\aleph _0, \end{aligned}$$
as S is \(\aleph _0\)-categorical. Hence \(S/\rho \) is \(\aleph _0\)-categorical. \(\square \)

If we drop the condition on Proposition 3.10 that the congruence is relatively automorphism preserving then the statement is no longer true. An example of an \(\aleph _0\)-categorical group with a non-\(\aleph _0\)-categorical quotient group is given by Rosenstein [35].

Corollary 3.11

Let S be an \(\aleph _0\)-categorical semigroup.
  1. (1)

    If \(\rho \) is a congruence preserved by automorphisms, then \(S/\rho \) is \(\aleph _0\)-categorical.

     
  2. (2)

    If S is inverse, then \(S/\sigma \) is an \(\aleph _0\)-categorical group.

     
  3. (3)

    The semigroup \(S/{{{\mathcal {H}}}}^{\flat }\) is \(\aleph _0\)-categorical, so that if S is inverse, then \(S/\mu \) is \(\aleph _0\)-categorical.

     
  4. (4)

    If \(\rho \) is a finitely generated congruence, then \(S/\rho \) is \(\aleph _0\)-categorical.

     
  5. (5)

    If I is an \({\overline{X}}\)-pivoted r.c. ideal of S for some finite tuple \({\overline{X}}\) of elements of S, then S / I is \(\aleph _0\)-categorical.

     

Proof

(1)–(3) follow from Example 3.5 and Proposition 3.10. For (4) we let \(\rho =\langle (u_1,v_1),\ldots ,(u_r,v_r)\rangle \) be a finitely generated congruence on S and let \({\overline{X}}=(u_1,v_1,\ldots , u_n,v_n)\). It is easy to see from the explicit description of \(\rho \) (see [19, Proposition 1.5.9]) that \(\rho \) is an \({\overline{X}}\)-pivoted r.a.p. congruence with pivot \({\overline{X}}\). The result is then immediate from Proposition 3.10.

For (5), suppose that I is an \({\overline{X}}\)-p.r.c. ideal of S. Let \(\phi \) be an automorphism of S which fixes \({\overline{X}}\), so that \(I\phi =I\) since I is an \({\overline{X}}\)-pivoted r.c. ideal. Then, for any \(a,b\in S\), we have
$$\begin{aligned} a \, \rho _I \, b \Leftrightarrow [a=b \text { or } a,b\in I] \Leftrightarrow [a\phi =b\phi \text { or } a\phi ,b\phi \in I] \Leftrightarrow a\phi \, \rho _I \, b\phi , \end{aligned}$$
thus showing that \(\rho _I\) is an \({\overline{X}}\)-r.a.p. congruence, and once more we call upon Proposition 3.10. \(\square \)

If S is a semigroup, then it is clear the intersection of all non-empty ideals is an ideal. If this is non-empty, it is denoted by K(S). Thus K(S), if it exists, is the unique minimum non-empty ideal of S.

Theorem 3.12

The principal factors of an \(\aleph _0\)-categorical semigroup S are \(\aleph _0\)-categorical, and either completely 0-simple, completely simple or null. Moreover, S has only finitely many principal factors, up to isomorphism.

Proof

For each \(a\in S\) let \(J(a)=S^1aS^1\) and \(I(a)=J(a){\setminus } J_a\). As in Corollary 3.4, \(\{(J(a),a):a\in S\}\) is a system of 1-pivoted p.r.c. subsemigroups of S and, since S is \(\aleph _0\)-categorical, the ideals J(a) are \(\aleph _0\)-categorical. Let \(\phi \) be an automorphism of S such that \(a\phi =b\), and so \(J(a)\phi =J(b)\). Moreover, as \({\mathcal {J}}\) is preserved by automorphisms we have \(J_a\phi =J_b\), and so
$$\begin{aligned} I(a)\phi = (J(a){\setminus } J_a)\phi =J(b){\setminus } J_b = I(b). \end{aligned}$$
Consequently, \(\{(I(a),a):a\in S\}\) is a system of 1-pivoted p.r.c. subsemigroups of S and, in particular, I(a) is an a-pivoted r.c. ideal of J(a) for each \(a\in S\). Hence J(a) / I(a) is \(\aleph _0\)-categorical by Corollary 3.11 (5). If K(S) exists, then it is a \({\mathcal {J}}\)-class of S, and is thus \(\aleph _0\)-categorical. Hence each principal factor of S is \(\aleph _0\)-categorical.

Moreover, as \(\phi |_{J(a)}\) is an isomorphism from J(a) to J(b) with \(I(a)\phi |_{J(a)}=I(b)\), it follows that the isomorphism \(\phi |_{J(a)}\) preserves \(\rho _{I(a)}\) and \(\rho _{I(b)}\), and so \(\phi \) induces an isomorphism from J(a) / I(a) to J(b) / I(b). Hence the set \(\{J(a)/I(a):a\in S\}\) of non-kernel principal factors of S has at most \(|S/\sim _{S,1}|\) elements, up to isomorphism. Since K(S) is unique, S has only finitely many principal factors, up to isomorphism.

By [5, Lemma 2.39], the principal factors of S are either 0-simple, simple or null. A periodic [0-]simple semigroup is completely [0-]simple (the result for 0-simple semigroups is given in [5, Corollary 2.56], from which the simple case follows). Hence as an \(\aleph _0\)-categorical semigroup is periodic by Corollary 2.3, each principal factor is either completely 0-simple, completely simple or null. \(\square \)

Recall that every null semigroup is \(\aleph _0\)-categorical by Example 2.7. To understand the \(\aleph _0\)-categoricity of an arbitrary semigroup it is therefore essential to examine the completely simple and completely 0-simple cases. The \(\aleph _0\)-categoricity of inverse completely 0-simple semigroups will be considered in Sect. 4, while the \(\aleph _0\)-categoricity of an arbitrary completely [0-]simple semigroup will be the main topic of a subsequent paper [29].

As we have seen in Proposition 2.11 and Corollary 3.11, the RNT is adept at dealing with a range of finiteness conditions. We end this section by studying a final finiteness condition: equivalence relations on a semigroup with finite equivalence classes.

Let S be a semigroup and \(\tau \) an equivalence relation on S. For each \(n\in {\mathbb {N}}\), define an equivalence relation \(\#_{S,\tau ,n}\) on \(S^n\) by \((a_1,\dots ,a_n) \, \#_{S,\tau ,n} \, (b_1,\dots ,b_n)\) if and only if there exists an automorphism \(\phi \) of S such that \((a_k\tau )\phi =b_k\tau \) for each \(1\le k \le n\).

Proposition 3.13

Let S be a semigroup and \(\tau \) an equivalence relation on S with each \(\tau \)-class being finite. Then \(|S^n/\#_{S,\tau ,n}|\) is finite for each \(n\in {\mathbb {N}}\) if and only if S is \(\aleph _0\)-categorical and \(A=\{|m\tau |:m\in S\}\) is finite.

Proof

Suppose that \(|S^n/\#_{S,\tau ,n}|\) is finite for each \(n\in {\mathbb {N}}\). Let \(Z=\{{\overline{a}}_i:i\in {\mathbb {N}}\}\) be an infinite set of n-tuples of S, where \({\overline{a}}_i=(a_{i1},\dots ,a_{in})\). Since \(|S^n/\#_{S,\tau ,n}|\) is finite, there exists an infinite subset \(\{{\overline{a}}_i:i\in I\}\) of Z such that \({\overline{a}}_i \, \#_{S,\tau ,n} \, {\overline{a}}_j\) for each \(i,j\in I\). Fix some \(p\in I\). Then for each \(i\in I\) there exists an automorphism \(\phi _i\) of S with \(({a}_{ik}\tau ) \phi _i={a}_{pk}\tau \) for each \(1 \le k \le n\). Hence \(a_{ik}\phi _i\in a_{pk}\tau \) for each \(1\le k \le n\), so that
$$\begin{aligned} {\overline{a}}_i\phi _i\in \{(z_1,\dots ,z_n):z_k\in a_{pk}\tau \}. \end{aligned}$$
Notice that the set \(\{(z_1,\dots ,z_n):z_k\in a_{pk}\tau \}\) is finite since each \(\tau \)-class is finite. Consequently, there exists distinct \(i,j\in I\) such that \({\overline{a}}_i\phi _i={\overline{a}}_j\phi _j\), so that \({\overline{a}}_i\phi _i\phi _j^{-1}={\overline{a}}_j\). Hence \({\overline{a}}_i \) and \({\overline{a}}_j\) are automorphically equivalent. It follows that S contains no infinite set of distinct n-automorphism types, and is thus \(\aleph _0\)-categorical by the RNT. Furthermore, by our usual argument we have that |A| is bound by \(|S / \#_{S,\tau ,1}|\).
Conversely, suppose S is \(\aleph _0\)-categorical and A is finite. Let \({\overline{m}}=(m_1,\dots ,m_n)\) and \({\overline{m}}'=(m_1',\dots ,m_n')\) be a pair of n-tuples of S, under the condition that \(|m_k\tau |=|m_k'\tau |\) for each k. Since each entry of an n-tuple of \(S^n\) has |A| potential cardinalities for its \(\tau \)-class, it follows that this condition has \(|A|^n\) choices. For each \(1\le k \le n\), let \(m_k\tau =\{a_{k1},\dots ,a_{ks_k}\}\) and \(m_{k}'\tau =\{b_{k1},\dots ,b_{ks_k}\}\), and let \(T(n)=s_1+s_2+\cdots + s_n\). Suppose further that
$$\begin{aligned} (a_{11},\dots ,a_{1s_1},a_{21},\dots ,a_{2s_2},\dots , a_{ns_n}) \, \sim _{S,T(n)} \, (b_{11},\dots ,b_{1s_1},b_{21},\dots ,b_{2s_2},\dots ,b_{ns_n}), \end{aligned}$$
via \(\phi \in \text {Aut}(S)\), say. Note that this condition also has finitely many choices as \(|S^{T(n)}/ \sim _{S,T(n)}|\) is finite for each \(n\in {\mathbb {N}}\) by the RNT. Moreover, \((m_k\tau )\phi =m_k'\tau \) for each k, since \(a_{kr}\phi =b_{kr}\) for each \(1\le r \le s_k\). Hence \({\overline{m}} \, \#_{S,\tau ,n} \, {\overline{m}}'\), and so \(|S^n/\#_{S,\tau ,n}|\) is finite by Lemma 2.8. \(\square \)

Corollary 3.14

Let S be a regular semigroup with each maximal subgroup being finite. Then S is \(\aleph _0\)-categorical if and only if \(|E(S)^n/\sim _{S,n}|\) is finite for each \(n\in {\mathbb {N}}\).

Proof

If S is \(\aleph _0\)-categorical, then
$$\begin{aligned} |E(S)^n/\sim _{S,n}|\le |S^n/\sim _{S,n}|<\aleph _0 \end{aligned}$$
for each \(n\in {\mathbb {N}}\) by the RNT.
Conversely, suppose \(|E(S)^n/\sim _{S,n}|\) is finite for each \(n\in {\mathbb {N}}\) and consider a pair of n-tuples of S given by \({\overline{a}}=({a_1},\dots ,{a_n})\) and \({\overline{b}}=({b_1},\,\dots ,{b_n})\). Since S is regular, there exists idempotents \(e_i,f_i,{\bar{e}}_i,{\bar{f}}_i\) of S with \(e_i \, {\mathcal {R}} \, a_i \, {\mathcal {L}} \, f_i\) and \({\bar{e}}_i \, {\mathcal {R}} \, b_i \, {\mathcal {L}} \, {\bar{f}}_i\) for each \(1\le i \le n\). Suppose further that
$$\begin{aligned} (e_1,f_1,e_2,f_2,\dots ,e_n,f_n) \, \sim _{S,2n} \, ({\bar{e}}_1,{\bar{f}}_1,{\bar{e}}_2,{\bar{f}}_2,\dots ,{\bar{e}}_n,{\bar{f}}_n), \end{aligned}$$
via \(\phi \in \text {Aut}(S)\), say. Then as \({\mathcal {R}}\) and \({\mathcal {L}}\) are automorphism preserving we have that \(R_{e_i}\phi =R_{{\bar{e}}_i}\) and \(L_{f_i}\phi =L_{{\bar{f}}_i}\) for each i, so that
$$\begin{aligned} H_{a_i}\phi =(R_{a_i}\cap L_{a_i})\phi = (R_{e_i}\cap L_{f_i})\phi =R_{e_i}\phi \cap L_{f_i}\phi = R_{{\bar{e}}_i}\cap L_{{\bar{f}}_i}=H_{b_i}. \end{aligned}$$
Hence \({\overline{a}} \, \#_{S,{\mathcal {H}},n} \, {\overline{b}}\), and we have thus shown that
$$\begin{aligned} |S^n/\#_{S,{\mathcal {H}},n}|\le |E(S)^{2n}/ \sim _{S,2n}| <\aleph _0. \end{aligned}$$
Since each maximal subgroup of S is finite, every \({\mathcal {H}}\)-class of S is finite by the regularity of S and [19, Lemma 2.2.3]. Hence S is \(\aleph _0\)-categorical by Proposition 3.13. \(\square \)

4 Building \(\aleph _0\)-categorical semigroups: Brandt semigroups, direct sums and 0-direct unions

In this section we consider the \(\aleph _0\)-categoricity of a number of well known constructions, the first being motivated by Theorem 3.12.

The Brandt semigroup over a groupGwith index setI, denoted \({\mathcal {B}}^0[G;I]\), is the set \((I\times G \times I)\cup \{0\}\) with multiplication \((i,g,j)0=0(i,g,j)=00=0\) and
$$\begin{aligned} (i,g,j)(k,h,l) = \left\{ \begin{array}{ll} (i,gh,l) &{}\quad \text {if } j=k,\\ 0 &{}\quad \text {if } j \ne k. \end{array} \right. \end{aligned}$$
Every Brandt semigroup is an inverse completely 0-simple semigroup and, conversely, an inverse completely 0-simple semigroup is isomorphic to some Brandt semigroup [19, Theorem 5.1.8]. Our early interest in Brandt semigroups from an \(\aleph _0\)-categorical perspective is due to the simplicity with which their automorphisms may be determined. The automorphism theorem for Brandt semigroups below is a direct consequence of [19, Theorem 3.4.1]:

Proposition 4.1

Let \(S={\mathcal {B}}^0[G;I]\) be a Brandt semigroup. Let \(\theta \) be an automorphism of G, and \(\pi \) a bijection of I. Then the map \(\psi :S\rightarrow S\) given by \(0\psi =0\) and \((i,g,j)\psi =(i\pi ,g\theta ,j\pi )\) for each \((i,g,j)\in S{\setminus } \{0\}\) is an automorphism, denoted \(\psi =(\theta ;\pi )\). Conversely, every automorphism of \({\mathcal {B}}^0[G;I]\) may be constructed in this manner.

Theorem 4.2

A Brandt semigroup \(S={\mathcal {B}}^0[G;I]\) is \(\aleph _0\)-categorical if and only if G is \(\aleph _0\)-categorical.

Proof

(\(\Rightarrow \)) Since G is isomorphic to each non-zero maximal subgroup \(B_i=\{(i,g,i):g\in G\}\) of S, the result follows from Corollary 3.7.

(\(\Leftarrow \)) By the RNT and Proposition 2.11, to prove the \(\aleph _0\)-categoricity of S it suffices to show that the number of n-automorphism types of \(S^*=S {\setminus } \{0\}\) is finite for each \(n\in {\mathbb {N}}\). Let \({\overline{a}}=((i_1,g_1,j_1),\dots , (i_n,g_n,j_n))\) and \({\overline{b}}=((k_1,h_1,\ell _1),\dots ,(k_n,h_n,\ell _n))\) be a pair of n-tuples of \(S^*\) such that
  1. (1)

    \((i_1,\dots , i_n,j_1,\dots , j_n) \, \natural _{I,2n} \, (k_1,\dots ,k_n,\ell _1,\dots ,\ell _n)\),

     
  2. (2)

    \((g_1,\dots ,g_n) \sim _{G,n} (h_1,\dots ,h_n)\), via \(\theta \in \text {Aut}(G)\), say.

     
By condition (1) there exists a bijection \(\pi \) from \(\{i_1,\dots ,i_n,j_1,\dots ,j_n\}\) to \(\{k_1,\dots ,k_n,\ell _1,\dots ,\ell _n\}\) given by \(i_r\pi =k_r\) and \(j_r\pi =\ell _r\) (\(1\le r \le n\)). Moreover, condition (1) has \(B_{2n}\) choices, which is finite. By the \(\aleph _0\)-categoricity of G, condition (2) also has finitely many choices. Take a bijection \({\bar{\pi }}\) of I which extends \(\pi \). Then \(\psi =(\theta ;{\bar{\pi }})\) is an automorphism of S by Proposition 4.1, and is such that
$$\begin{aligned} (i_r,g_r,j_r)\psi =(i_r{\bar{\pi }},g_r\theta ,j_r{\bar{\pi }})=(i_r\pi , h_r,j_r\pi )=(k_r,h_r,\ell _r) \end{aligned}$$
for each \(1\le r \le n\). Hence \({\overline{a}} \, \sim _{S,n} \, {\overline{b}}\), and so S is \(\aleph _0\)-categorical by Lemma 2.8. \(\square \)

The classification of \(\aleph _0\)-categorical Brandt semigroups is an example of building \(\aleph _0\)-categorical semigroups from \(\aleph _0\)-categorical ‘ingredients’, in this case groups (and sets). The rest of the article is attributed to investigating constructions of this form for a number of rudimentary examples, the next being direct sums.

Let I be an indexing set and suppose that for each \(i\in I\) we have a monoid \(M_i\) with identity \(1_i\). By the direct sum\(S=\bigoplus _{i\in I}M_i\) we mean the submonoid
$$\begin{aligned} \{ (m_i)_{i\in I}: m_i=1_i \text{ for } \text{ all } \text{ but } \text{ finitely } \text{ many } i\in I\} \end{aligned}$$
of the direct product \(P=\Pi _{i\in I}M_i\). Rosenstein [34] showed that any group that is a direct sum of copies of finitely many finite groups is \(\aleph _0\)-categorical if and only if every group which occurs infinitely often in the sum is abelian. For our purposes we extend his result slightly as follows. Recall that the centreZ(G) of a group G is the subgroup of G consisting of all those \(a\in G\) such that \(ag=ga\) for all \(g\in G\).

Lemma 4.3

(cf. [34, Theorem 3]) Let \(S=\bigoplus _{i\in {\mathbb {N}}}M_i\) where each \(M_i\) is a finite group. Then if S is \(\aleph _0\)-categorical, all but finitely many of the \(M_i\)’s are abelian.

Proof

Suppose for a contradiction that
$$\begin{aligned} M_{i_1}, M_{i_2}, \ldots \end{aligned}$$
are all non-abelian. Pick \(a_{j}\in M_{i_j}\) with \(a_{j}\notin Z(M_{i_j})\), so that \(|\{b^{-1}a_jb:b\in M_{i_j}\}|=m_j>1\). Let \(\overline{s_j}\in S\) be defined by
$$\begin{aligned} (\overline{s_j})_{i_k}=a_k \text{ for } 1\le k\le j, (\overline{s_j})_i=1_i \text{ else }. \end{aligned}$$
Then \(|\{{\overline{b}}^{-1}\overline{s_j}{\overline{b}}:{\overline{b}}\in S\}|=m_1 m_2 \ldots m_j\) so that for \(i\ne j\) we cannot have \((\overline{s_i})\sim _{S,1}(\overline{s_j})\), contradicting the \(\aleph _0\)-categoricity of S. \(\square \)

Theorem 4.4

Let \(S=\bigoplus _{i\in {\mathbb {N}}}M_i\) be a direct sum of finite monoids \(M_i\). Then S is \(\aleph _0\)-categorical if and only if S is a direct product of a finite monoid and an abelian group of bounded order.

Proof

Suppose first that S is \(\aleph _0\)-categorical. We first show that all but finitely many of the monoids \(M_i\) are groups. Suppose we have an infinite sequence
$$\begin{aligned} M_{i_1}, M_{i_2}, \ldots \end{aligned}$$
such that each \(M_{i_j}\) is not a group. By this hypothesis, for each \(j\in {{\mathbb {N}}}\) we may choose a non-identity idempotent \(e_j\in M_{i_j}\) such that \(e_j\) is maximal in \(E(M_{i_j}){\setminus }\{ 1_{i_j}\}\) (under the natural partial order given by \(e \, \le \, f\) if and only if \(ef=fe=e\)).
Consider the sequence
$$\begin{aligned} \overline{s_1}, \overline{s_2},\overline{s_3},\ldots \end{aligned}$$
where
$$\begin{aligned} (\overline{s_j})_{i_k}=e_k \text{ for } 1\le k\le j, (\overline{s_j})_i=1_i \text{ else }. \end{aligned}$$
For each \(j\in {{\mathbb {N}}}\) there are exactly \(2^j\) idempotents of S equal to or greater than \(\overline{s_j}\) in the natural partial order, so that the elements \(\overline{s_j}\) lie in distinct \(\sim _{S,1}\)-classes, a contradiction. Thus \(S=M\times G\) where M is a finite monoid and G is a direct sum of finite groups.

The group of units of S is \(\aleph _0\)-categorical by Corollary 3.7, and is a direct sum of G and the group of units H of M. By Lemma 4.3 all but finitely many of the constituents of the direct sum forming G are abelian, so that \(G=K\times W\) where K is finite and W is an abelian group of bounded order, hence \(\aleph _0\)-categorical by [34, Theorem 2]. Thus \(S=M\times K\times W\) where \(M\times K\) is finite and W is an abelian group of bounded order.

The converse is clear as \(\aleph _0\)-categoricity is preserved by finite direct products by [16]. \(\square \)

To translate to the semigroup case requires some care, as here the direct sum does not embed into the direct product. Let \(M_i\) be a semigroup for each \(i\in I\). By the direct sum of the semigroups \(M_i\) (\(i\in I\)) we mean the semigroup
$$\begin{aligned} S=\left\langle \bigcup _{i\in I} \underline{M_i} : \underline{m_i}\, \underline{m_i'}=\underline{m_im'_i},\, \underline{m_i}\, \underline{m_j}=\underline{m_j}\, \underline{m_i}\, \forall i\ne j, m_i,m_i'\in M_i,m_j\in M_j\right\rangle , \end{aligned}$$
where \(\underline{M_i}=\{ \underline{m_i}:m_i\in M_i\}\) for all \(i\in I\).

Proposition 4.5

Let S be the direct sum of the finite semigroups \(M_i\) (\(i\in I\)). Then S is \(\aleph _0\)-categorical if and only if I (and hence S) is finite.

Proof

Suppose that S is \(\aleph _0\)-categorical. For each \(M_i\) we choose a maximal idempotent \(e_i\). If I is infinite, then without loss of generality we may take \(I={{\mathbb {N}}}\). Let \(\underline{s_i}=\underline{e_1} \, \underline{e_2}\ldots \underline{e_i}\). Notice that for each \(\underline{s_i}\) there are precisely \(2^i-1\) idempotents greater than or equal to \(\underline{s_i}\), so that each \(\underline{s_i}\) lies in a distinct \(\sim _{S,1}\)-class. Thus I is finite. The converse is immediate by Corollary 2.1. \(\square \)

Given the disappointing nature of Proposition 4.5 we focus attention on a different construction, which yields useful results. The basic definitions and results are taken from [3].

A semigroup with zero S is a 0-direct union or orthogonal sum of the subsemigroups \(S_i\) (\(i\in A\)) with zero, if the following hold:
  1. (1)

    \(S_i\ne \{0\}\) for each \(i\in A\);

     
  2. (2)

    \(S=\bigcup _{i\in A} S_i\);

     
  3. (3)

    \(S_i\cap S_j=S_iS_j=\{0\}\) for each \(i\ne j\).

     
We denote S as \(\bigsqcup _{i\in A}^0 S_i\). The family \({\mathcal {S}}=\{S_i:i\in A\}\) is called a 0-direct decomposition of S, and the \(S_i\) are called the summands of S. Note that each summand of S forms an ideal of S. If \({\mathcal {S}}\) and \({\mathcal {S}}'\) are a pair of 0-direct decompositions of S, then we say that \({\mathcal {S}}\) is greater than\({\mathcal {S}}'\) if each member of \({\mathcal {S}}\) is a subsemigroup of some member of \({\mathcal {S}}'\). We say that S is 0-directly indecomposable if \(\{S\}\) is the unique 0-direct decomposition of S.

Example 4.6

Let \({\mathcal {B}}={\mathcal {B}}^0[G;I]\) be a Brandt semigroup, and consider the group with zero \(B^0_i=\{(i,g,i):g\in G\}\cup \{0\}\) for each \(i\in I\). Then \(B_i^0B_j^0=\{0\}\) if \(i\ne j\), and so \(\bigcup _{i\in I}B_i^0\) forms a 0-direct union of the subsemigroups \(B_i^0\). Note that if \(|I|>1\) then \(\bigcup _{i\in I}B_i^0\) forms a proper subsemigroup of \({\mathcal {B}}\), since it does not contain the element (igj) for any \(i\ne j\) and \(g\in G\).

A subset T of a semigroup S is consistent if, for \(x,y\in S\), \(xy\in T\) implies that \(x,y\in T\). A subset T of a semigroup with zero is 0-consistent if \(T{\setminus } \{0\}\) is consistent. The integral connection between 0-consistency and 0-direct decompositions is that a semigroup with zero S is 0-directly indecomposable if and only if S has no proper 0-consistent ideals [3, Lemma 4]. Consequently, every completely 0-simple semigroup is 0-directly indecomposable.

The central result of [3] was proving that that every semigroup with zero has a greatest 0-direct decomposition, and that the summands of such a decomposition are precisely the 0-directly indecomposable ideals. The importance of the existence of a greatest 0-direct decomposition for \(\aleph _0\)-categoricity is highlighted in the following proposition.

Proposition 4.7

Let S be a semigroup with zero and let \({\mathcal {S}}= \{S_i:i\in A\}\) be the greatest 0-direct decomposition of S. Let \(\pi :A\rightarrow A\) be a bijection and \(\phi _i:S_i\rightarrow S_{i\pi }\) an isomorphism for each \(i \in A\). Then the map \(\phi :S \rightarrow S\) given by
$$\begin{aligned} s_i\phi = s_i\phi _i \quad (s_i\in S_i) \end{aligned}$$
is an automorphism of S, denoted \(\phi =\bigsqcup ^0_{i\in A} \phi _i\). Moreover, every automorphism of S can be constructed in this way.

Proof

Let \(\phi \) be constructed as in the hypothesis of the proposition. Since \(0\phi _i=0\) for each \(i\in A\) the map is well-defined, and it is clearly bijective. Let \(a\in S_i\) and \(b\in S_j\). If \(i=j\) then
$$\begin{aligned} (ab)\phi = (ab)\phi _i=(a\phi _i)(b\phi _i)=(a\phi )(b\phi ), \end{aligned}$$
and if \(i\ne j\) then
$$\begin{aligned} (ab)\phi = 0\phi = 0 = (a\phi _i)(b\phi _j) =(a\phi )(b\phi ). \end{aligned}$$
Hence \(\phi \) is an isomorphism.
Conversely, if \(\phi '\) is an automorphism of S, then
$$\begin{aligned} {\mathcal {S}}\phi '=\{S_i\phi ':i\in A\} \end{aligned}$$
is clearly a 0-direct decomposition of S. For each summand \(S_i\) there exists \(k\in A\) such that \(S_i\subseteq S_k\phi '\) since \({\mathcal {S}}\) is the greatest 0-direct decomposition. If \(S_i\subseteq S_k\phi '\cap S_{k'}\phi '\) then \(S_i=\{0\}\) as \({\mathcal {S}}\phi '\) is a 0-direct decomposition of S, a contradiction. Hence the element k is unique. On the other hand, if \(S_i,S_j\subseteq S_k\phi '\), then \(S_i\phi '^{-1},S_j\phi '^{-1}\subseteq S_k\), and so as \(\{S_i\phi '^{-1}:i\in A\}\) is also a 0-direct decomposition of S, we have that \(i=j\) since \(S_k\) is 0-direct indecomposable. Hence there exists a bijection \(\pi '\) of A such that \(S_{i}\phi '=S_{i\pi '}\) for each \(i\in A\) as required. \(\square \)

Theorem 4.8

Let S be a semigroup with zero and let \({\mathcal {S}}= \{S_i:i\in A\}\) be the greatest 0-direct decomposition of S. Then S is \(\aleph _0\)-categorical if and only if each \(S_i\) is \(\aleph _0\)-categorical and \({\mathcal {S}}\) is finite, up to isomorphism.

Proof

It follows immediately from Proposition 4.7 that \(\{(S_i,x_i):i\in A\}\) forms a system of 1-pivoted p.r.c subsemigroups of S for any \(x_i\in S_i^*=S_i{\setminus } \{0\}\). Hence if S is \(\aleph _0\)-categorical then each \(S_i\) is \(\aleph _0\)-categorical and \({\mathcal {S}}\) is finite, up to isomorphism, by Proposition 3.3.

For the converse direction, let each summand be \(\aleph _0\)-categorical and suppose there exists exactly \(r\in {\mathbb {N}}\) summands, up to isomorphism. Let \(S_{\rho _1},\dots , S_{\rho _r}\) be representatives of the isomorphism types of the summands of S and, for each \(\mu \in A\), let \(\phi _{\mu }\) be an isomorphism from \(S_\mu \) to its unique isomorphic representative in \(S_{\rho _1},\dots , S_{\rho _r}\). By Proposition 2.11 it suffices to show that the number of n-automorphism types of \(S^*=S{\setminus } \{0\}\) is finite for each \(n\in {\mathbb {N}}\). Let \({\overline{a}}=(a_1,\dots ,a_n)\) and \({\overline{b}}=(b_1,\dots ,b_n)\) be a pair of n-tuples of \(S^*\) with \(a_k\in S_{i_k}\) and \(b_k\in S_{j_k}\) for each \(1\le k \le n\), say. Impose the condition that \(a_i,a_j\) belong to the same summand if and only if \(b_i,b_j\) belong to the same summand, for each \(1\le i, j\le n\). This is clearly equivalent to the map \(\pi :\{i_1,\dots ,i_n\} \rightarrow \{j_1,\dots ,j_n\}\) given by \(i_k\pi =j_k\) being a bijection, and thus the number of choices for this condition is equal to \(B_n\). Suppose also that \(S_{i_k}\cong S_{j_k}\) for each k, noting that this condition has \(r^n\) choices. For each \(1\le s \le r\), let \(a_{s1},\dots ,a_{sn_s}\) be precisely the entries of \({\overline{a}}\) which are elements of summands isomorphic to \(S_{\rho _s}\), noting that the same is true of \(b_{s1},\dots ,b_{sn_s}\) by our second condition. Note also that \(\{1,\dots ,n\}=\{11,\dots ,1n_1,21,\dots ,2n_2,\dots ,rn_r\}\). We impose a final condition on our pair of n-tuples which forces, for each \(1\le s \le r\),
$$\begin{aligned} (a_{s1}\phi _{i_{s1}},\dots ,a_{sn_s}\phi _{i_{sn_s}}) \, \sim _{S_{\rho _s},n_s} \, (b_{s1}\phi _{j_{s1}},\dots ,b_{sn_s}\phi _{j_{sn_s}}) \end{aligned}$$
via \(\psi _s\in \text {Aut}(S_{\rho _s})\), say (where if \(n_s=0\) then we take any automorphism of \(S_{\rho _s}\)). By the \(\aleph _0\)-categoricity of each \(S_{\rho _s}\) this condition also has finitely many choices. For each \(1 \le s \le r\) and \(1\le t \le n_s\) we have that \(\phi _{i_{st}}\psi _s \phi _{j_{st}}^{-1}\) is an isomorphism from \(S_{i_{st}}\) to \(S_{j_{st}}\), and is such that
$$\begin{aligned} a_{st}\phi _{i_{st}}\psi _s \phi _{j_{st}}^{-1} = b_{st}. \end{aligned}$$
Let \({\pi }'\) be a bijection of A which extends \(\pi \) and which preserves the isomorphism types of the summands, so that \(S_i\cong S_{i\pi '}\). Such a bijection exists since each \(S_{i_k}\) is isomorphic to \(S_{j_k}\). For each \(i\in A{\setminus } \{i_1,\dots , i_n\}\), let \(\Psi _i\) be an isomorphism from \(S_i\) to \(S_{i\pi '}\), and we let \(\Psi _{i_{st}}=\phi _{i_{st}}\psi _s \phi _{j_{st}}^{-1}\) for each \(1 \le s \le r\) and \(1\le t \le n_s\). Then \(\Psi =\bigsqcup ^0_{i\in A} \Psi _i\) is an automorphism of S by Proposition 4.7, and is such that \({\overline{a}}\Psi ={\overline{b}}\) since \(\Psi \) extends each \(\phi _{i_{st}}\psi _s \phi _{j_{st}}^{-1}\). Since each of our conditions has only finitely many choices, \((S^*)^n\) has only finitely many n-automorphism types (over S) by Lemma 2.8, and thus S is \(\aleph _0\)-categorical. \(\square \)

When studying \(\aleph _0\)-categorical semigroups with zero, it therefore suffices to examine 0-directly indecomposable semigroups.

We observe that without the condition of \({\mathcal {S}}\) being the greatest 0-direct decomposition of S, the converse direction of Theorem 4.8 need not be true. For example, for each \(n\in {\mathbb {N}}\), let \(N_n\) be a null semigroup on n non-zero elements. Then \(N=\bigsqcup ^0_{i\in {\mathbb {N}}} N_i\) is a countably infinite null semigroup, and is thus \(\aleph _0\)-categorical by Example 2.7. However the set of summands of N is not finite, up to isomorphism.

A semigroup S with zero is called primitive if each of its non-zero idempotents is primitive. It follows from the work of Hall in [17] that a regular semigroup S is primitive if and only if S is isomorphic to a 0-direct union of completely 0-simple semigroups. Since each completely 0-simple semigroup is 0-directly indecomposable, we obtain the following immediate consequence to Theorem 4.8.

Corollary 4.9

Let \(S_i\) (\(i\in A\)) be a collection of completely 0-simple semigroups. Then \(\bigsqcup _{i\in A}^0 S_i\) is \(\aleph _0\)-categorical if and only if each \(S_i\) is \(\aleph _0\)-categorical and \(\{S_i:i\in A\}\) is finite, up to isomorphism.

A classification of primitive regular \(\aleph _0\)-categorical semigroups via its completely 0-simple semigroup ideals then follows. In particular, by Proposition 4.1 and Theorem 4.2 we have the following classification of primitive inverse semigroups:

Corollary 4.10

A primitive inverse semigroup S is \(\aleph _0\)-categorical if and only if S is isomorphic to \(\bigsqcup _{i\in A}^0 {\mathcal {B}}^0[G_i;I_i]\) such that each \(G_i\) is \(\aleph _0\)-categorical, \(\{G_i:i\in A\}\) is finite up to isomorphism, and \(\{|I_i|:i\in A\}\) is finite.

5 Building \(\aleph _0\)-categorical semigroups: Semidirect products and \({\mathcal {P}}\)-semigroups

Given that \(\aleph _0\)-categoricity has been shown by Grzegorczyk to be inherited by finite direct products [16], the next natural question is to assess semidirect products; in this section we do so in the case of a semigroup acting on a finite semigroup. Our work requires the following variant of \(\aleph _0\)-categoricity, and the subsequent pair of lemmas:

Definition 5.1

Given a semigroup S and a collection \({\mathcal {A}}=\{S_i:i\in A\}\) of subsets of S, we let Aut\((S;{\mathcal {A}})\) denote the group of automorphisms of S which fix each \(S_i\) (\(i\in A\)) setwise. We call S\(\aleph _0\)-categorical over\({\mathcal {A}}\) if Aut\((S;{\mathcal {A}})\) has finitely many orbits on its action on \(S^n\) for each \(n\in {\mathbb {N}}\). We let \(\sim _{S,{\mathcal {A}},n}\) denote the resulting equivalence relation on \(S^n\).

With notation as above, Definition 5.1 is equivalent to the structure consisting of the semigroup S together with a collection of unary relations corresponding to the subsets \(S_i\) (\(i\in I\)), being \(\aleph _0\)-categorical. Moreover, if \({\overline{X}}=(x_1,\dots ,x_t)\) is a tuple of elements of S, then the condition that S is \(\aleph _0\)-categorical over \({\overline{X}}\) is equivalent to S being \(\aleph _0\)-categorical over \(\{\{x_1\},\dots ,\{x_t\}\}\).

Lemma 5.2

Let S be a semigroup with a system of t-pivoted p.r.c. subsets \(\{(S_i,{\overline{X}}_i):i\in I\}\). Then S is \(\aleph _0\)-categorical over \({\mathcal {A}}=\{S_i:i\in I\}\) if and only if S is \(\aleph _0\)-categorical and \({\mathcal {A}}\) is finite.

Proof

If S is \(\aleph _0\)-categorical over \({\mathcal {A}}\), then trivially S is \(\aleph _0\)-categorical. Suppose \(i,j\in I\) are such that \({\overline{X}}_i \, \sim _{S,{\mathcal {A}},t} \, {\overline{X}}_j\), via \(\phi \in \text {Aut}(S;{\mathcal {A}})\), say. Then \(S_i\phi =S_i\), while \(S_i\phi =S_j\) since \(\{(S_i,{\overline{X}}_i):i\in I\}\) is a system of t-pivoted p.r.c. subsets of S. Hence \(S_i=S_j\), and so the cardinality of \({\mathcal {A}}\) is bound by the number of t-automorphism types over \({\mathcal {A}}\).

Conversely, suppose S is \(\aleph _0\)-categorical with \({\mathcal {A}}\) finite, say \({\mathcal {A}}=\{S_1,\dots , S_r\}\). Let \({\overline{a}}=(a_1,\dots ,a_n)\) and \({\overline{b}}=(b_1,\dots ,b_n)\) be a pair of n-tuples of S such that \(({\overline{a}},{\overline{X}}_1,\dots ,{\overline{X}}_r) \, \sim _{S,n+rt} \, ({\overline{b}},{\overline{X}}_1,\dots ,{\overline{X}}_r)\), via \(\psi \in \text {Aut}(S)\), say. Then as each pivot is fixed by \(\psi \), the sets \(S_i\) are fixed setwise by \(\psi \), so that \(\psi \in \text {Aut}(S;{\mathcal {A}})\). Hence as \({\overline{a}}\psi ={\overline{b}}\) we have that
$$\begin{aligned} |S^n/\sim _{S,{\mathcal {A}},n}|\le |S^{n+rt}/\sim _{S,n+rt}|<\aleph _0 \end{aligned}$$
and so S is \(\aleph _0\)-categorical over \({\mathcal {A}}\). \(\square \)

A simple adaptation of the proof of the lemma above also gives:

Lemma 5.3

Let S be a semigroup, let \(t,r\in {{\mathbb {N}}}\), and for each \(k\in \{ 1,\ldots , r\}\) let \({\overline{X}}_k\in S^t\). Suppose also that \(S_k\) is an \({\overline{X}}_k\)-pivoted relatively characteristic subset of S for \(1\le k \le r\). Then S is \(\aleph _0\)-categorical if and only if S is \(\aleph _0\)-categorical over \(\{S_1,\dots ,S_r\}\).

Now suppose S is a semigroup acted on (on the left) by a monoid T via endomorphisms. That is, we have a map \(T\times S\rightarrow S\) denoted by \((t,s)\mapsto t\cdot s\), such that for all \(t,t'\in T\) and \(s,s'\in S\) we have \(tt'\cdot s=t\cdot (t'\cdot s)\), \(1\cdot s=s\) and \(t\cdot (ss')=(t\cdot s)(t\cdot s')\). We may then construct a semigroup on the set \(S\times T\) with binary operation \((s,t)(s',t')=(s(t\cdot s'),tt')\). The resulting semigroup is denoted by \(S\rtimes T\), and is called a semidirect product of S by T. We refer the reader to [40] for an introduction to monoid actions.

Given a semidirect product \(S\rtimes T\), we define a relation \(\kappa \) on T by
$$\begin{aligned} t \, \kappa \, t' \Leftrightarrow s(t\cdot s') = s(t' \cdot s') \,\, (\forall s,s'\in S). \end{aligned}$$
(5.1)
Then \(\kappa \) is clearly an equivalence relation on T, and if S is finite then \(T/\kappa \) is finite. If S is a monoid and \(t\cdot 1_S=1_S\) for all \(t\in T\), then we say that T acts monoidally; note in this case \(S\rtimes T\) is a monoid, and the definition of \(\kappa \) simplifies to
$$\begin{aligned} t \, \kappa \, t' \Leftrightarrow t\cdot s = t' \cdot s \,\, (\forall s\in S). \end{aligned}$$

Theorem 5.4

Let \(M=S\rtimes T\) be a semidirect product of S and T, where S is finite. If T is \(\aleph _0\)-categorical over \(T/\kappa \), then M is \(\aleph _0\)-categorical.

The converse holds if S is a monoid with trivial group of units and T acts monoidally, or if S is a semilattice.

Proof

Suppose that T is \(\aleph _0\)-categorical over \(T/\kappa \). Let \({\overline{a}}=((s_1,t_1),\dots ,(s_n,t_n))\) and \({\overline{b}}=((s_1',t_1'),\dots ,(s_n',t_n'))\) be n-tuples of M under the conditions that
  1. (1)

    \(s_k=s_k'\) for each \(1\le k \le n\),

     
  2. (2)

    \((t_1,\dots ,t_n) \, \sim _{T,T/\kappa ,n} \, (t_1',\dots ,t_n')\) via \(\theta \in \text {Aut}(T;T/\kappa )\), say.

     
Note that the first condition has \(|S|^n\) choices. We claim that the bijection \(\phi :M\rightarrow M\) given by \((s,t)\phi =(s,t\theta )\) is an automorphism of M. Given \((s,t),(s',t')\in M\),
$$\begin{aligned} ((s,t)(s',t'))\phi&= (s(t\cdot s'),tt')\phi = (s(t \cdot s'),(tt')\theta )= (s(t\theta \cdot s'),(tt')\theta )\\&= (s(t\theta \cdot s'),(t\theta )(t'\theta )) = (s,t\theta )(s',t'\theta ) = (s,t)\phi (s',t')\phi , \end{aligned}$$
where the third equality is due to \(t \, \kappa \, t\theta \), so in particular \(s(t \cdot s') = s(t\theta \cdot s')\). Hence \(\phi \) is indeed an automorphism of M. Moreover, for each \(1 \le k \le n\),
$$\begin{aligned} (s_k,t_k)\phi = (s_k,t_k\theta ) = (s_k',t_k') \end{aligned}$$
and so \({\overline{a}} \phi = {\overline{b}}\). We therefore have that
$$\begin{aligned} |M^n/\sim _{M,n}| \le |S|^n \cdot | T^n/\sim _{T,T/\kappa ,n}| < \aleph _0 \end{aligned}$$
since S is finite and T is \(\aleph _0\)-categorical over \(T/\kappa \). Hence M is \(\aleph _0\)-categorical by the RNT.
Conversely, suppose that M is \(\aleph _0\)-categorical. Enumerate the elements of S as \(\{ s_1,\ldots , s_r\}\). Let \(s=1_S\) if S is a monoid and let \(s=0\) where 0 is the least idempotent of S if S is a semilattice. Let \({\overline{t}}=(t_1,\ldots , t_n)\) and \({\overline{u}}=(u_1,\ldots , u_n)\) be n-tuples of T under the conditions that
  1. (1)

    \(t_k \, \kappa \, u_k\) for each \(1\le k \le n\),

     
  2. (2)

    \(\big ((s_1,1),\ldots , (s_r,1), (s,t_1), \ldots ,(s,t_n)\big ) \sim _{M,n+r} \, \big ((s_1,1),\ldots , (s_r,1), (s,u_1), \ldots , (s,u_n)\big )\) via \(\theta \in {\text {Aut}}(M)\), say.

     
Recalling that s is fixed, for any \(t\in T\) we define \(t\phi \) to be the second coordinate of \((s,t)\theta \). We claim that \(\phi \in {\text {Aut}}(T)\) and preserves \(T/\kappa \).
Case (i):Sa monoid with trivial group of units, so\(s=1_S\). We first show that \((1_S,t)\theta =(1_S,t\phi )\). By definition, we have \((1_S,t)\theta =(s',t\phi )\) for some \(s'\in S\). Choose \((b,w)\in S\rtimes T\) such that \((b,w)\theta =(1_S, t\phi )\). Then \(\theta \) fixes \((s',1)\) by condition (2), so that
$$\begin{aligned} (1_S,t)\theta =(s',t\phi )=(s',1)(1_S, t\phi )=(s',1)\theta (b,w)\theta = ((s',1)(b,w))\theta =(s'b,w)\theta . \end{aligned}$$
Hence \(1_S=s'b\), giving \(s'=1_S\) as \(H_{1_S}\) is trivial and S is finite (giving that an element with a left inverse lies in \(H_{1_S}\)).
We have thus shown that \((1_S,t)\theta =(1_S, t\phi )\), whence it follows that for any \(u\in S, t\in T\) we have
$$\begin{aligned} (u,t)\theta =\big ((u,1)(1_S,t)\big )\theta =(u,1)(1_S,t\phi )=(u,t\phi ). \end{aligned}$$
It is now easy to see that \(\phi \in {\text {Aut}}(T)\), since T acts monoidally.
Case (ii):Sa semilattice, so\(s=0\). We first show that \((0,t)\theta =(0,t\phi )\). To see this, let \((0,t)\theta =(0',t\phi )\); making use of (2) we have
$$\begin{aligned} (0',t\phi )=(0,t)\theta =\big ((0,1)(0,t)\big )\theta = (0,1)(0',t\phi )=(0,t\phi ) \end{aligned}$$
giving \(0'=0\). It is now easy to see that \(\phi \) yields an automorphism of T.
Let \(e\in S\) and \(t\in T\) and suppose that \((e,t)\theta =(e',t')\). Then
$$\begin{aligned} (0,t')=(0,1)(e',t')=(0,1)\theta (e,t)\theta =\big ((0,1)(e,t)\big )\theta =(0,t)\theta =(0,t\phi ), \end{aligned}$$
so that \(t'=t\phi \). Let \(u\in S\) with \(u>0\). We may then suppose for induction that for all \(v\in S\) with \(u>v\) and for all \(t\in T\) we have \((v,t)\theta =(v,t\phi )\). Then with \((u,t)\theta =(u',t\phi )\) we have
$$\begin{aligned} (u',t\phi )=(u,t)\theta =\big ((u,1)(u,t)\big )\theta =(u,1)(u',t\phi )=(uu',t\phi ), \end{aligned}$$
so that \(u'=uu'\) and \(u'\le u\). If \(u'<u\) we are led to the contradiction that \((u',t)\theta =(u',t\phi )=(u,t)\theta \). Thus \(u'=u\) and we deduce that for any \(w\in S,t\in T\) we have \((w,t)\theta =(w,t\phi )\).

In each case, for any \(u,u'\in S\) and \(t\in T\), by applying \(\theta \) to the product \((u,t)(u',1)\) we immediately see that \(t\,\kappa \, t\phi \). Moreover, as \((s,t_i)\theta =(s,u_i)\) we have \(t_i\phi =u_i\) for \(1\le i\le n\). Thus T is \(\aleph _0\)-categorical over \(T/\kappa \). \(\square \)

Open Problem 5.5

Can we weaken the conditions on S in the converse to Theorem 5.4?

Example 5.6

Let T be a semigroup acting trivially on a finite semigroup S, so that \(t\cdot s = s\) for each \(s\in S, t\in T\). It follows that \(\kappa \) is the universal relation. Hence \(S\rtimes T\) is \(\aleph _0\)-categorical if T is \(\aleph _0\)-categorical over \(T/\kappa = \{T\}\), which is clearly equivalent to T being \(\aleph _0\)-categorical. Note that \(S\rtimes T\) is simply the direct product of S and T, and so we recover Grzegorczyk’s result [16].

Example 5.7

Let \(L=\{x_1,\dots ,x_r\}\) be a finite left zero band and \(S=\bigsqcup _{i\in I}^0 S_i\) an \(\aleph _0\)-categorical 0-direct union of 0-directly indecomposable \(S_i\). Then as \(L^0=L\cup \{0\}\) is 0-directly indecomposable and \(\aleph _0\)-categorical, it follows from Theorem 4.8 that \(S'=S\sqcup ^0 L^0\) is also \(\aleph _0\)-categorical. Let \(S'\) act on its ideal \(L^0\) by left multiplication, so that \(t\cdot s = ts\) for each \(t\in S'\) and \(s\in L^0\). Then this is an action by endomorphisms as
$$\begin{aligned} t\cdot (s_1s_2) = ts_1s_2 = \left\{ \begin{array}{ll} t &{}\quad \text {if } t\in L^0\\ 0 &{}\quad \text {else } \end{array} \right. = (ts_1)(ts_2). \end{aligned}$$
We aim to show that \(L^0 \rtimes S'\) is \(\aleph _0\)-categorical. Notice that \(t \, \kappa \, t'\) if and only if \(sts'=st's'\) for all \(s,s'\in L^0\). However,
$$\begin{aligned} sts' = \left\{ \begin{array}{ll} s &{}\quad \text {if } t\in L,\\ 0 &{}\quad \text {else } \end{array} \right. \end{aligned}$$
and it follows that the \(\kappa \)-classes are L and \(S'{\setminus } L\). Since any automorphism of \(S'\) which fixes L setwise clearly fixes \(S'{\setminus } L\) setwise, we have that \(S'\) is \(\aleph _0\)-categorical over \(S'/\kappa \) if and only if \(S'\) is \(\aleph _0\)-categorical over \(\{L\}\). From Lemma 2.6, we have that \(S'\) is \(\aleph _0\)-categorical over \((x_1,\dots ,x_r)\), hence over \(\{L\}\), and so over \(S'/\kappa \). Hence \(L^0 \rtimes S'\) is \(\aleph _0\)-categorical by Theorem 5.4.

Our final example comes from studying the semidirect product of a group and a semilattice. Such semigroups are examples of E-unitary inverse semigroups, a class that plays a central role in the study of inverse semigroups. A semigroup S is E-unitary if whenever \(e,es\in E(S)\) then \(s\in E(S)\). In the case of inverse semigroups, this condition is equivalent to \({\mathcal {R}}\cap \sigma =\iota \) (or, indeed, to \({\mathcal {L}}\cap \sigma =\iota \)), where \(\sigma \) is the the congruence defined in Example 3.5 (2), and \(\iota \) is the equality relation. This equivalent condition is often referred to as that of being proper. McAlister [22, 23] showed that every inverse semigroup has an E-unitary cover (a pre-image via an idempotent separating morphism) and characterised the structure of E-unitary semigroups via what are known as \({\mathcal {P}}\)-semigroups. The construction of a \({\mathcal {P}}\)-semigroup, which can be found in [19, Chapter 5], is very close to that of a semidirect product of a semilattice by a group. Indeed, a semidirect product of a semilattice by a group is E-unitary, and every \({\mathcal {P}}\)-semigroup embeds into such a semidirect product [26]. However, a free inverse semigroup provides an example of a \({\mathcal {P}}\)-semigroup which is not a semidirect product of a semilattice by a group.

Let \({\mathcal {X}}\) be a partially ordered set with order \(\le \), and let \({\mathcal {Y}}\) be an order ideal of \({{{\mathcal {X}}}}\) which forms a semilattice under \(\le \). Let G be a group which acts on \({\mathcal {X}}\) by order automorphisms, and suppose in addition that
  1. (1)

    \(G{\mathcal {Y}}={\mathcal {X}}\) and

     
  2. (2)

    \(g{\mathcal {Y}}\cap {\mathcal {Y}}\ne \emptyset \) for all \(g\in G\).

     
Then the triple \((G,{\mathcal {X}},{\mathcal {Y}})\) is called a McAlister triple. We may then take
$$\begin{aligned} {\mathcal {P}}={\mathcal {P}}(G,{{{\mathcal {X}}}},{{{\mathcal {Y}}}}) = \{(A,g)\in {{{\mathcal {Y}}}}\times G: g^{-1}A\in {{\mathcal {Y}}}\} \end{aligned}$$
and define an operation on \({\mathcal {P}}\) by the rule
$$\begin{aligned} (A,g)(B,h)=(A\wedge gB,gh). \end{aligned}$$

Theorem 5.8

[23] Let \((G,{\mathcal {X}},{\mathcal {Y}})\) be a McAlister triple. Then \( {\mathcal {P}}={\mathcal {P}}(G,{{{\mathcal {X}}}},{{{\mathcal {Y}}}})\) is an E-unitary inverse semigroup with semilattice of idempotents \(E({\mathcal {P}})={\mathcal {Y}}\times \{ 1\}\) isomorphic to \({{\mathcal {Y}}}\). Moreover, any E-unitary inverse semigroup S is isomorphic to some \({\mathcal {P}}(G,{\mathcal {X}},{\mathcal {Y}})\) where \(G=S/\sigma \) and \({\mathcal {Y}}=E(S)\).

The semigroup \({\mathcal {P}}(G,{{\mathcal {X}}},{{\mathcal {Y}}})\) is often referred to as a \({\mathcal {P}}\)-semigroup. Notice that if \({{\mathcal {X}}}={{\mathcal {Y}}}\) then \({\mathcal {P}}(G,{{\mathcal {X}}},{{\mathcal {Y}}})={{\mathcal {Y}}}\rtimes G\). One can show that for any \((A,g),(B,h)\in {\mathcal {P}}(G,{{\mathcal {X}}},{{\mathcal {Y}}})\) we have
$$\begin{aligned} (A,g)\,{\mathcal {R}}\, (B,h)\Leftrightarrow A=B \text{ and } (A,g)\,{{{\mathcal {L}}}}\, (B,h)\Leftrightarrow g^{-1}A=h^{-1}B. \end{aligned}$$
For any \({\mathcal {P}}\)-semigroup \({\mathcal {P}}={\mathcal {P}}(G,{{\mathcal {X}}},{{\mathcal {Y}}})\) we have \({\mathcal {P}}/\sigma \cong G\), and so the \(\aleph _0\)-categoricity of \({\mathcal {P}}\) passes to G by Corollary 3.11 (2). Our aim is therefore to consider when the converse holds, or rather, what conditions on G force \({\mathcal {P}}\) to be \(\aleph _0\)-categorical? We require McAlister’s [23] description of morphisms between \({\mathcal {P}}\)-semigroups, which simplifies to automorphisms as follows.

Proposition 5.9

Let \({\mathcal {P}}={\mathcal {P}}(G,{{{\mathcal {X}}}},{{{\mathcal {Y}}}})\) be a \({\mathcal {P}}\)-semigroup. Let \(\theta \in {\text {Aut}}(G)\) and \(\psi :{{\mathcal {X}}}\rightarrow {{\mathcal {X}}}\) be an order-automorphism such that \(\psi |_{{\mathcal {Y}}}\in {\text {Aut}}({{\mathcal {Y}}})\). Suppose also that, for all \(g\in G\) and \(A\in {{\mathcal {X}}}\),
$$\begin{aligned} (gA)\psi =(g\theta )(A\psi ). \end{aligned}$$
(5.2)
Then the map \(\phi :{\mathcal {P}}\rightarrow {\mathcal {P}}\) given by \((A,g)\phi =(A\psi ,g\theta )\) is an automorphism, denoted \(\phi =(\psi ;\theta )\). Conversely, every automorphism of \({\mathcal {P}}\) is of this type.
Notice that if \(\psi \) is the identity on \({{\mathcal {X}}}\), then for (5.2) to hold the automorphism \(\theta \) of G must preserve the group action, that is,
$$\begin{aligned} gA=(g\theta )A \quad (\forall g\in G)(\forall A\in {\mathcal {X}}). \end{aligned}$$
(5.3)
On the other hand, if \(\theta \) is the identity on G, then for (5.2) to hold \(\psi \) must be a G-act morphism, in addition to being an order-automorphism, that is,
$$\begin{aligned} (gA)\psi =g(A\psi ). \end{aligned}$$
(5.4)
We first consider the case of finite \({{\mathcal {Y}}}\). We recall from Sect. 3 that we denote by \(\mu \) the largest idempotent separating congruence on an inverse semigroup S, and that \(\mu \) is also precisely the largest congruence contained in \({{{\mathcal {H}}}}\). From the formula for \(\mu \) on [19, p.160], it is clear that \(\mu \) has finitely many classes if and only if E(S) is finite, so that if \({\mathcal {P}}={\mathcal {P}}(G,{{{\mathcal {X}}}},{{{\mathcal {Y}}}})\), then \({\mathcal {P}}/\mu \) is finite if and only if \({{\mathcal {Y}}}\) is finite.
For each \(A\in {{\mathcal {Y}}}\) let \(T_A\subseteq G\) be defined by
$$\begin{aligned} T_A=\{ g\in G: g^{-1}A\in {{\mathcal {Y}}}\}, \end{aligned}$$
noting that \(G=\bigcup _{A\in {\mathcal {Y}}} T_A\) since \(g{{\mathcal {Y}}}\cap {{\mathcal {Y}}}\ne \emptyset \) for each \(g\in G\). We also define a relation \(\sim _A\) on \(T_A\) by:
$$\begin{aligned} g\sim _A h\Leftrightarrow g^{-1}U=h^{-1}U \text{ for } \text{ all } U\le A. \end{aligned}$$
Clearly each \(\sim _A\) is an equivalence.

The following lemma is immediate from [19, p.217].

Lemma 5.10

Let \({\mathcal {P}}={\mathcal {P}}(G,{{{\mathcal {X}}}},{{{\mathcal {Y}}}})\) be a \({\mathcal {P}}\)-semigroup. Then
$$\begin{aligned} (A,g)\,\mu \, (B,h)\Leftrightarrow A=B \text{ and } g\sim _A h. \end{aligned}$$

Remark 5.11

The finiteness of \({{\mathcal {Y}}}\) in \({\mathcal {P}}={\mathcal {P}}(G,{{{\mathcal {X}}}},{{{\mathcal {Y}}}})\) does not imply that \({{\mathcal {X}}}\) is finite.

To see this, consider the semilattice \({{\mathcal {X}}}\) on \(\{ X_i:i\in {\mathbb {N}}^0\}\cup \{0\}\) where 0 is a least element and the remaining elements are atoms. Let G be the free group on \(\{ g_i:i\in {\mathbb {N}}\}\). Each \(g_i\) determines an order-automorphism of \({{\mathcal {X}}}\) by setting \(g_iX_0=X_i\), \(g_iX_i=X_0\) and \(g_iX_j=X_j\) for all \(j\ne 0,i\). Since G is free on \(\{ g_i:i\in {\mathbb {N}}\}\), we may lift the action of the generators to an action of G on \({{\mathcal {X}}}\) by order-automorphisms. Setting \({{\mathcal {Y}}}=\{ X_0,0\}\) we see that \((G,{{{\mathcal {X}}}},{{{\mathcal {Y}}}})\) is a McAlister triple.

Theorem 5.12

Let \({\mathcal {P}}={\mathcal {P}}(G,{{{\mathcal {X}}}},{{{\mathcal {Y}}}})\) be a \({\mathcal {P}}\)-semigroup such that \({{\mathcal {Y}}}\) is finite. Then \({\mathcal {P}}\) is \(\aleph _0\)-categorical if and only if G is \(\aleph _0\)-categorical over \(\bigcup _{A\in {\mathcal {Y}}}T_A/\sim _A\).

Proof

Let \({\mathcal {A}}=\bigcup _{A\in {\mathcal {Y}}}T_A/\sim _A\). Let \({{\mathcal {Y}}}=\{ Y_1,\ldots , Y_k\}\) and for each \(Y_i\in {{\mathcal {Y}}}\) pick a set of representatives
$$\begin{aligned} (Y_i,g_1^i),\ldots , (Y_i,g_{n(i)}^i) \end{aligned}$$
of the \(\mu \)-classes sitting in the \({\mathcal {R}}\)-class of \((Y_i,1)\), so that \(g_1^i,\ldots , g_{n(i)}^i\) are representatives of the \(\sim _{Y_i}\)-classes.
\((\Rightarrow )\) Let \((a_1,\dots ,a_n)\) and \((b_1,\dots ,b_n)\) be a pair of n-tuples of G. Then for any j with \( 1\le j\le n\), as \(a_j^{-1} {\mathcal {Y}} \cap {\mathcal {Y}} \ne \emptyset \) there exists \(A_j\in {\mathcal {Y}}\) such that \((A_j,a_j)\in {\mathcal {P}}\); similarly form \((B_j,b_j)\in {\mathcal {P}}\). Impose the condition that
$$\begin{aligned} \big ((Y_1,g^1_1),\ldots , (Y_1,g^1_{n(1)}), \ldots , (Y_k,g^k_1),\ldots , (Y_k,g^k_{n(k)}),(A_1,a_1),\ldots \dots ,(A_n,a_n)\big ) \end{aligned}$$
and
$$\begin{aligned} \big ((Y_1,g^1_1),\ldots , (Y_1,g^1_{n(1)}), \ldots , (Y_k,g^k_1),\ldots , (Y_k,g^k_{n(k)} ),(B_1,b_1),\dots ,(B_n,b_n)\big ) \end{aligned}$$
are related by \(\, \sim _{{\mathcal {P}},t+n} \, \) where \(t=|{\mathcal {P}}/\mu |\), via \(\phi =(\psi ;\theta )\in \text {Aut}({\mathcal {P}})\), say. Clearly each \(\mu \)-class is fixed by \(\phi \) and in particular each \(A\in {{\mathcal {Y}}}\) is fixed by \(\psi \). Suppose that \(g\in T_A\). Then
$$\begin{aligned} g^{-1}A=(g^{-1}A)\psi =(g^{-1}\theta ) (A\psi )=(g^{-1}\theta ) A=(g\theta )^{-1}A \end{aligned}$$
so that \(g\theta \in T_A\). Moreover, for any \(U\le A\) we have \(g^{-1}U\in {{\mathcal {Y}}}\) and
$$\begin{aligned} g^{-1}U=(g^{-1}U)\psi =(g\theta )^{-1}(U\psi )=(g\theta )^{-1}U, \end{aligned}$$
so that \(g\sim _A g\theta \). Thus \(\theta \in {\text {Aut}}(G)\) and \(\theta \) preserves \({\mathcal {A}}\). Clearly \(a_i\theta =b_i\) for \(1\le i\le n\). Hence G is \(\aleph _0\)-categorical over \({\mathcal {A}}\).

\((\Leftarrow )\) Since \({\mathcal {Y}}\) is finite and G is \(\aleph _0\)-categorical over \({\mathcal {A}}\), from any infinite list of n-tuples of elements of \({\mathcal {P}}\) we may pick out a pair \({\overline{a}}=((A_1,g_1),\dots ,(A_n,g_n))\) and \({\overline{b}}=((A_1,h_1),\dots ,(A_n,h_n))\) such that \((g_1,\dots ,g_n) \, \sim _{G,{\mathcal {A}},n} \, (h_1,\dots ,h_n)\), via some \(\theta \in \text {Aut}(G;{\mathcal {A}})\). Define \(\phi :{\mathcal {P}}\rightarrow {\mathcal {P}}\) by \((A,g)\phi =(A,g\theta )\). Let \(g\in G\) and \(A,B\in {{\mathcal {Y}}}\) with \(g^{-1}A\in {{\mathcal {Y}}}\). Since \(g\in T_A \) and \(A\wedge gB\) exists and is less than A, we have \(g^{-1}(A\wedge gB)=(g\theta )^{-1}(A \wedge gB)\). Also, \(A\wedge gB=gg^{-1}(A\wedge gB)=g(g^{-1}A\wedge B)=g((g\theta )^{-1}A\wedge B)\). It follows that \(A\wedge gB=A\wedge (g\theta )B\) so that, consequently, \(\phi \) is an isomorphism. Hence \({\mathcal {P}}\) is \(\aleph _0\)-categorical. \(\square \)

Given a \({\mathcal {P}}\)-semigroup \({\mathcal {P}}={\mathcal {P}}(G,{{{\mathcal {X}}}},{{{\mathcal {Y}}}})\), we define a relation \(\nu \) on G by
$$\begin{aligned} g \, \nu \, h \Leftrightarrow gA = hA \,\, (\forall A\in {{\mathcal {X}}}). \end{aligned}$$
(5.5)
Then \(\nu \) is clearly an equivalence relation, and if \(g \, \nu \, h\) then \(g \, \sim _A \, h\) for any \(A\in {{\mathcal {Y}}}\), with \(g\in T_A\) if and only if \(h\in T_A\). Hence if \(g\in G\) then \(g\nu \) is a subset of an element of \(T_A/ \sim _A\) for some \(A\in {{\mathcal {Y}}}\). It follows that if G is \(\aleph _0\)-categorical over \(G/\nu \), then G is \(\aleph _0\)-categorical over \({\mathcal {A}}\). If \({\mathcal {X}}\) is finite, then there are only finitely many \(\nu \)-classes. Lemma 5.10 and Theorem 5.12 thus prove the reverse direction to the following result:

Theorem 5.13

Let \({\mathcal {P}}={\mathcal {P}}(G,{{{\mathcal {X}}}},{{{\mathcal {Y}}}})\) be a \({\mathcal {P}}\)-semigroup such that \({\mathcal {X}}\) is finite. Then \({\mathcal {P}}\) is \(\aleph _0\)-categorical if and only if G is \(\aleph _0\)-categorical over \(G/\nu \).

Proof

Suppose \({\mathcal {P}}\) is \(\aleph _0\)-categorical and \({{\mathcal {X}}}=\{X_1,\dots ,X_r\}\) is finite. For each \(i\in \{ 1,\ldots , r\}\) choose and fix \(g_i\in G\) and \(Y_i\in {{\mathcal {Y}}}\) such that \(X_i=g_iY_i\). Let \({\overline{g}}=(a_1,\dots ,a_n)\) and \({\overline{h}}=(b_1,\dots ,b_n)\) be a pair of n-tuples of G. For each \(1\le j \le n\) and \(1\le i \le r\), form \((A_j,a_j),(B_j,b_j),(C_i,g_i)\in {\mathcal {P}}\) for some \(A_j,B_j,C_i\in {{\mathcal {Y}}}\). Impose the conditions that
$$\begin{aligned} \big ( (A_1,a_1),\dots , (A_n,a_n),(C_1,g_1),\dots ,(C_r,g_r),(Y_1,1),\dots ,(Y_r,1)\big ) \end{aligned}$$
and
$$\begin{aligned} \big ( (B_1,b_1),\dots , (B_n,b_n),(C_1,g_1),\dots ,(C_r,g_r),(Y_1,1),\dots ,(Y_r,1)\big ) \end{aligned}$$
are related by \(\sim _{{\mathcal {P}},n+2r}\), via \(\phi =(\psi ;\theta )\in \text {Aut}({\mathcal {P}})\). Then
$$\begin{aligned} X_i\psi =(g_iY_i)\psi =(g_i\theta ) (Y_i\psi ) = g_i Y_i=X_i \end{aligned}$$
for each i, and so \(\psi =I_{{{\mathcal {X}}}}\). For any \(g\in G\) and \(X_i\in {{\mathcal {X}}}\) we then have
$$\begin{aligned} gX_i = (gX_i)\psi = (g\theta )(X_i\psi ) = (g\theta ) X_i, \end{aligned}$$
so that \(g \, \nu \, g\theta \). Thus \(\theta \in \text {Aut}(G)\) preserves \(G/\nu \), and is such that \(a_j\theta =b_j\) for \(1\le j \le n\). Hence G is \(\aleph _0\)-categorical over \(G/\nu \). \(\square \)

We handle the corresponding case where G is finite by introducing a new structure. In the definition of a \({\mathcal {P}}\)-semigroup \({\mathcal {P}}={\mathcal {P}}(G,{{{\mathcal {X}}}},{{{\mathcal {Y}}}})\) we have that G acts on the poset \({\mathcal {X}}\) by order automorphisms. Thus we may refer to \({{\mathcal {X}}}\) as a G-poset [11] (the partial order on G being equality). We consider the structure obtained by augmenting the signature for G-posets with a unary relation \({\mathcal {Z}}\) and, suppressing \(\le \) for convenience, refer to the new structure as the augmentedG-poset\(({{\mathcal {X}}},{\mathcal {Z}})\). Thus \(({{\mathcal {X}}},{\mathcal {Z}})\) has universe \({{\mathcal {X}}}\) and signature \((\le , {\mathcal {Z}}, \lambda _g: g\in G)\) where \(\lambda _g\) is the action of g on \({{\mathcal {X}}}\). An automorphism \(\psi \) of \(({{\mathcal {X}}},{\mathcal {Z}})\) must therefore be a G-act isomorphism (that is, \( (gX)\psi =g(X\psi )\) for all \(g\in G, X\in {{\mathcal {X}}}\)), in addition to being an order automorphism of \({{\mathcal {X}}}\) fixing \({\mathcal {Z}}\) setwise.

Theorem 5.14

Let \({\mathcal {P}}={\mathcal {P}}(G,{{{\mathcal {X}}}},{{{\mathcal {Y}}}})\) be a \({\mathcal {P}}\)-semigroup such that G is finite. Then \({\mathcal {P}}\) is \(\aleph _0\)-categorical if and only if the augmented G-poset \(({{\mathcal {X}}},{{\mathcal {Y}}})\) is \(\aleph _0\)-categorical.

Proof

Let \(G=\{ g_1,\ldots , g_m\}\) and pick \(D_j\in {\mathcal {Y}}\) with \((D_j,g_j)\in {\mathcal {P}}\), for \(1\le j\le m\).

Suppose first that \({\mathcal {P}}\) is \(\aleph _0\)-categorical. For any \(n\in {{\mathbb {N}}}\) and infinite sequence of n-tuples of \({{\mathcal {X}}}\), we can find \(h_1,\ldots , h_n\in G\) and a subsequence in which every n-tuple can be written as \((X_1,\ldots ,X_n)\), where \(X_i\in h_i{{\mathcal {Y}}}\) for all \(1\le i \le n\). From the \(\aleph _0\)-categoricity of \({\mathcal {P}}\) we can find distinct elements \((h_1Y_1,\ldots ,h_nY_n)\) and \((h_1Z_1,\ldots ,h_nZ_n)\) of our sequence (where \(Y_i,Z_i\in {{\mathcal {Y}}}\) for \(1\le i\le n\)) such that
$$\begin{aligned}&\big ((D_1,g_1),\ldots , (D_m,g_m),(Y_1,1),\ldots , (Y_n,1)\big )\\&\quad \sim _{{\mathcal {P}},m+n} \big ((D_1,g_1), \ldots ,(D_m,g_m),(Z_1,1),\ldots , (Z_n,1)\big ) \end{aligned}$$
via \(\phi =(\psi ;\theta )\), where clearly \(\theta =I_G\). Then for any \(g\in G, X\in {{\mathcal {X}}}\) we have
$$\begin{aligned} (gA)\psi =(g\theta )( A\psi )=g(A\psi ) \end{aligned}$$
so that \(\psi \) is a G-act isomorphism, in addition to possessing the properties that \(\psi \in {\text {Aut}}({{\mathcal {X}}})\) and \(\psi |_{{\mathcal {Y}}}\in {\text {Aut}}({{\mathcal {Y}}})\). Thus \(\psi \) is an automorphism of the augmented G-act \(({{\mathcal {X}}},{{\mathcal {Y}}})\). Moreover, we have
$$\begin{aligned} (h_iY_i)\psi =h_i (Y_i\psi )=h_iZ_i \end{aligned}$$
so that
$$\begin{aligned} (h_1Y_1, \ldots , h_nY_n)\sim _{({\mathcal {X}},{\mathcal {Y}}), n} (h_1Z_1, \ldots , h_nZ_n) \end{aligned}$$
as required. Thus \(({\mathcal {X}},{\mathcal {Y}})\) is \(\aleph _0\)-categorical.
Conversely, suppose that \(({\mathcal {X}},{\mathcal {Y}})\) is \(\aleph _0\)-categorical and we have an infinite sequence of n-tuples of \({\mathcal {P}}\). Since G is finite and \(({\mathcal {X}},{\mathcal {Y}})\) is \(\aleph _0\)-categorical we may find a distinct pair \(\big ((A_1,h_1),\ldots , (A_n,h_n)\big )\) and \(\big ((B_1,h_1),\ldots , (B_n,h_n)\big )\) such that
$$\begin{aligned} (A_1,\ldots , A_n)\sim _{({\mathcal {X}},{\mathcal {Y}}),n} (B_1,\ldots , B_n) \end{aligned}$$
via \(\psi \). As \(\psi \) is a G-act isomorphism, it is immediate that \((\psi ;I_G)\) is in \({\text {Aut}}({\mathcal {P}})\) and moreover,
$$\begin{aligned} \big ((A_1,h_1),\ldots , (A_n,h_n)\big )\sim _{{\mathcal {P}},n}\big ((B_1,h_1),\ldots , (B_n,h_n)\big ) \end{aligned}$$
via \((\psi ;I_G)\). Thus \({\mathcal {P}}\) is \(\aleph _0\)-categorical as required. \(\square \)

To deal with the case of a \({\mathcal {P}}\)-semigroup where both the semilattice \({{\mathcal {Y}}}\) and group G are infinite, we require a little more sophistication. In the sequel to this article we obtain classes of \(\aleph _0\)-categorical E-unitary semigroups with infinite semilattice of idempotents, by restricting our attention to those with central idempotents.

Footnotes

  1. 1.

    We thank Prof. John Truss of the University of Leeds for bringing this example to our attention.

Notes

Acknowledgements

This work forms part of the Ph.D. of the second author at the University of York, funded by EPSRC. The authors would also like to thank the referee for his/her very careful reading of their manuscript, and the very detailed and helpful comments provided, which have resulted in a much improved article.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of YorkYorkUK
  2. 2.Institut für AlgebraTechnische Universität DresdenDresdenGermany

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