$\aleph_0$-categoricity of semigroups II

A countable semigroup is $\aleph_0$-categorical if it can be characterised, up to isomorphism, by its first order properties. In this paper we continue our investigation into the $\aleph_0$-categoricity of semigroups. Our main results are a complete classification of $\aleph_0$-categorical orthodox completely 0-simple semigroups, and descriptions of the $\aleph_0$-categorical members of certain classes of strong semilattices of semigroups.


Introduction
A countable structure is ℵ 0 -categorical if it is uniquely determined by its first order properties, up to isomorphism. The concept of ℵ 0 -categoricity arises naturally from model theory, however it has a purely algebraic formulation, as we explain in Section 2. Significant results exist for both relational and algebraic structures from the point of view of ℵ 0categoricity, but, until recently, little was known in the context of semigroups. This article is the second of a pair initiating and developing the study of ℵ 0 -categorical semigroups. For background and motivation we refer the reader to [10] and our first article [5].
We explore in [5] the behaviour of ℵ 0 -categoricity with respect to standard constructions, such as quotients and subsemigroups. For example, ℵ 0 -categoricity of a semigroup is inherited by both its maximal subgroups and its principal factors. Differences with the known theory for groups and rings emerged, for example, any ℵ 0 -categorical nil ring is nilpotent, but the same is not true for semigroups. While keeping the machinery at a low level, we were able to give, amongst other results, complete classifications of ℵ 0 -categorical primitive inverse semigroups and of E-unitary inverse semigroups with finite semilattices of idempotents.
For the work in this current article, it is helpful to develop some general strategies and then apply them in various contexts. In view of this, in Section 2, we introduce ℵ 0categoricity in the setting of (first order) structures. Although we will mostly be working in the context of semigroups, this broader view will be useful for studying certain structures, such as graphs and semilattices, which naturally arise in our considerations of semigroups. Key results from [5] are given in this setting. In particular, we formalise the previously defined concept of ℵ 0 -categoricity over a set of subsets; the ℵ 0 -categoricity of rectangular bands over any set of subrectangular bands acts as a useful example.
In Section 3 we construct a handy method for dealing with the ℵ 0 -categoricity of semigroups in which their automorphisms can be built from certain ingredients. This is then used in Section 4 to study the ℵ 0 -categoricity of strong semilattices of semigroups. The main results of this article are in Section 5, where we continue from [5] our study into the ℵ 0 -categoricity of completely 0-simple semigroups. We follow a method of Graham and Houghton by considering graphs arising from Rees matrix semigroups, which necessitated our study of ℵ 0 -categoricity in the general setting of structures.
We shall assume that all structures considered will be of countable cardinality.
2. The ℵ 0 -categoricity of a structure We begin by translating a number of results in [5] to the general setting of (first order) structures. Their proofs easily generalize, and as such we shall omit them, referencing only the corresonding result in [5].
A (first order) structure is a set M together with a collection of constants C, finitary relations R, and finitary functions F defined on M . We denote the structure as (M ; R, F, C), or simply M where no confusion may arise. Each constant element is associated with a constant symbol, each n-ary relation of M is associated with an n-ary relational symbol, and each n-ary function is associated with an n-ary function symbol. The collection L of these symbols is called the signature of M . We follow the usual convention of not distinguishing between the constants/relations/functions of M , and their corresponding abstract symbols in L.
Our main example is that of a semigroup (S, ·), where S is a set together with a single (associative) binary operation · , and so the associated signature consists of a single binary function symbol.
A property of a structure is first order if it can be formulated within first order predicate calculus. A (countable) structure is ℵ 0 -categorical if it can be uniquely classified by its first order properties, up to isomorphism.
Given a structure M , we say that a pair of n-tuples a = (a 1 , . . . , a n ) and b = (b 1 , . . . , b n ) of M are automorphically equivalent or belong to the same n-automorphism type if there exists an automorphism φ of M such that aφ = b, that is, a i φ = b i for each i ∈ {1, . . . , n}. We denote this equivalence relation as a ∼ M,n b. We call Aut(M ) oligomorphic if Aut(M ) has only finitely many orbits in its action on M n for each n ≥ 1, that is, if each |M n / ∼ M,n | is finite. The central result in the study of ℵ 0 -categorical structures is the Ryll-Nardzewski Theorem, which translates the concept to the study of oligomorphic automorphism groups (see [10]): An immediate consequence of the RNT is that any characteristic substructure inherits ℵ 0 -categoricity, where a subset is called characteristic if it is invariant under automorphisms of the structure. However, key subsemigroups of a semigroup such as maximal subgroups and principal ideals are not necessarily characteristic, and a more general definition is required: Definition 2.2. [5, Definition 3.1] Let M be a structure and, for some fixed t ∈ N, let {X i : i ∈ I} be a collection of t-tuples of M . Let {A i : i ∈ I} be a collection of subsets of M with the property that for any automorphism φ of M such that there exists i, j ∈ I with X i φ = X j , then φ| A i is a bijection from A i onto A j . Then we call A = {(A i , X i ) : i ∈ I} a system of t-pivoted pairwise relatively characteristic (t-pivoted p.r.c.) subsets (or, substructure, if each A i is a substructure) of M . The t-tuple X i is called the pivot of A i (i ∈ I). If |I| = 1 then, letting A 1 = A and X 1 = X, we write {(A, X)} simply as (A, X), and call A an X-pivoted relatively characteristic (X-pivoted r.c.) subset/substructure of M . Definition 2.2 was shown to be of use in regard to, for example, Green's relations. In particular {(H e , e) : e ∈ E(S)} was shown to form a system of 1-pivoted p.r.c. subgroups of a semigroup S. It then follows from Proposition 2.3 that maximal subgroups inherit ℵ 0 -categoricity, and moreover there exists only finitely many non-isomorphic maximal subgroups in an ℵ 0 -categorical semigroup.
We use the RNT in conjunction with [5,Lemma 2.8] to prove that a structure M is ℵ 0categorical in the following way. For each n ∈ N, let γ 1 , . . . , γ r be a finite list of equivalence relations on M n such that M n /γ i is finite for each 1 ≤ i ≤ r and γ 1 ∩ γ 2 ∩ · · · ∩ γ r ⊆ ∼ M,n .
A consequence of the two aforementioned results is that M is ℵ 0 -categorical. A condition imposed on n-tuples of M will naturally translate to an equivalence relation, and we will say that a condition has finitely many choices if its corresponding equivalence relation has finitely many equivalence classes.
Example 2.4. Recalling [5, Example 2.10], consider the equivalence X,n on n-tuples of a set X given by A pair of n-tuples a and b are X,n -equivalent if and only if there exists a bijection φ : {a 1 , . . . , a n } → {b 1 , . . . , b n } such that a i φ = b i , and the number of X,n -classes of X n is finite, for each n ∈ N. Note also that if M is a structure then any pair of n-automorphically equivalent tuples are clearly M,n -equivalent.
Let M be a structure and A = {A i : i ∈ I} a collection of subsets of M . We may extend the signature of M to include the unary relations A i (i ∈ I). We denote the resulting structure asM = (M ; A), which we call a set extension of M . If A = {A 1 , . . . , A n } is finite, then we may simply writeM as (M ; A 1 , . . . , A n ).
Notice that automorphisms ofM are simply those automorphisms of M which fix each M i setwise, that is automorphisms φ such that A i φ = A i (i ∈ I). The set of all such automorphisms will be denoted Aut(M ; A), and clearly forms a subgroup of Aut(M ). The ℵ 0 -categoricity ofM is therefore equivalent to our previous notion of M being ℵ 0 -categorical over A in [5].
Lemma 2.5. [5, Lemma 6.2] Let M be a structure with a system of t-pivoted p.r.c. subsets Lemma 2.6. [5, Lemma 6.3] Let M be a structure, let t, r ∈ N, and for each 1 ≤ k ≤ r let X k ∈ M t . Suppose also that A k is an X k -pivoted relatively characteristic subset of M Consequently, if S is an ℵ 0 -categorical semigroup and G 1 , . . . , G n is a collection of maximal subgroups of S then (S; However, note that not every ℵ 0 -categorical set extension of a semigroup requires the subsets to be relatively characteristic. We claim that any set extension of a rectangular band by a finite set of subrectangular bands is ℵ 0 -categorical. This result is of particular use in the next section when considering the ℵ 0 -categoricity of normal bands.
Recall that every rectangular band can be written as a direct product of a left zero and right zero semigroup. The following isomorphism theorem for rectangular bands will be vital for proving our claim, and follows immediately from [13,Corollary 4.4
Theorem 2.8. If B is a rectangular band and B 1 , . . . , B r is a finite list of subrectangular bands of B, thenB = (B; B 1 , . . . , B r ) is ℵ 0 -categorical. In particular, a rectangular band is ℵ 0 -categorical.
Proof. Let B = L × R, where L is a left zero semigroup and R is a right zero semigroup. For each 1 ≤ k ≤ r, let Define a pair of equivalence relations σ L and σ R on L and R, respectively, by The equivalence classes of σ L are simply the set L \ 1≤k≤r B L k together with certain intersections of the sets B L k . Since r is finite, it follows that L/σ L is finite, and similarly R/σ R is finite. Let a = ((i 1 , j 1 ), . . . , (i n , j n )) and b = ((k 1 , 1 ), . . . , (k n , n )) be a pair of n-tuples of B under the four conditions that (1) i s σ L k s for each 1 ≤ s ≤ n, (2) j s σ R s for each 1 ≤ s ≤ n, (3) (i 1 , . . . , i n ) L,n (k 1 , . . . , k n ), (4) (j 1 , . . . , j n ) R,n ( 1 , . . . , n ), where L,n and R,n are the equivalence relations given by (2.1). By conditions (3) and (4), there exists bijections given by i s φ L = k s and j s φ R = s for each 1 ≤ s ≤ n. By condition (1), we can pick a bijection Φ L of L which extends φ L and fixes each σ L -classes setwise, and similarly construct Hence there exists ∈ L and r ∈ R such that (iΦ L , r) and ( , jΦ R ) are in B k , so that and Φ −1 R setwise fixing the σ L -classes and σ R -classes, respectively. Following our previous argument we have B k Φ −1 ⊆ B k , and so B k Φ = B k for each k. Thus Φ is an automorphism ofB, and is such that for each 1 ≤ s ≤ n, so that a ∼B ,n b. Hence, as each of the four conditions on a and b have finitely many choices, it follows thatB is ℵ 0 -categorical.
Note that any set can be considered as a structure with no relations, functions or constants. Every bijection of the set is therefore an automorphism, and as such all sets are easily shown to be ℵ 0 -categorical. In fact a simplification of the proof of Theorem 2.8 gives:  For many of the structures we will consider, automorphisms can be built from isomorphisms between their components. For example, for a strong semilattice of semigroups S = [Y ; S α ; ψ α,β ], we can construct automorphisms of S from certain isomorphisms between the semigroups S α . In this example we also require an automorphism of the semilattice Y , which acts as an indexing set for the semigroups S α . We now extend this idea by setting up some formal machinery to deal with structures in which the automorphisms are built from a collection of data.   x ∈ M i ∩ M j then by taking π to be the identity map ofN , we have that xφ i , xφ j ∈ M i ∩ M j for all φ i ∈ Aut(M i ) and φ j ∈ Aut(M j ) by Condition (3.4). However, for our work the sets M i will mostly be pairwise disjoint, or will all intersect at an element which is fixed by every isomorphism between the M i . For example, M could be a semigroup containing a zero, and 0 is the intersection of each of the sets M i .
Note also that no link needs to exist between the signatures L and K. For most of our examples they will be the signature of semigroups and the signature of sets (the empty signature), respectively.
Given an (M ; M ;N ; Ψ)-system A = {M i : i ∈ N } in M , we aim to show that, if N is ℵ 0 -categorical and each M i possess a stronger notion of ℵ 0 -categoricity, then M is ℵ 0categorical. The stronger notion that we require comes from the following definition, which generalises the notion of ℵ 0 -categoricity of set extensions. Definition 3.3. Let M be a structure and Ψ a subgroup of Aut(M ). Then we say that that M is ℵ 0 -categorical over Ψ if Ψ has only finitely many orbits in its action on M n for each n ≥ 1. We denote the resulting equivalence relation on M n as ∼ M,Ψ,n .
By taking Ψ to be those automorphisms which fix certain subsets of M we recover our original definition of ℵ 0 -categoricity of a set extension. Similarly, by taking Ψ to be those automorphisms which preserve a fixed equivalence relation, or those which fix certain equivalence classes, we obtain a pair of notions defined in [5]. Proof. LetN = (N ; N 1 , . . . , N r ) and, for each 1 ≤ k ≤ r, fix some m k ∈ N k . For each i ∈ N k , let θ i ∈ Ψ i,m k , noting that such an element exists by Condition (3.1) on Ψ. Let a = (a 1 , . . . , a n ) and b = (b 1 , . . . , b n ) be a pair of n-tuples of M , with a t ∈ M it and b t ∈ M jt , and such that (i 1 , . . . , i n ) ∼N ,n (j 1 , . . . , j n ) via π ∈ Aut(N ), say. For each 1 ≤ k ≤ r, let i k1 , i k2 , . . . , i kn k be the entries of (i 1 , . . . , i n ) belonging to N k , where k1 < k2 < · · · < kn k , and set a k = (a k1 , . . . , a kn k ) ∈ (M ) n k .
We similarly form each b k , observing that as i t π = j t for each 1 ≤ t ≤ n and π fixes the sets N j setwise (1 ≤ j ≤ r) the elements j k1 , j k2 , . . . , j kn k are precisely the entries of (j 1 , . . . , j n ) belonging to N k , so that b k = (b k1 , . . . , b kn k ) for some b kt ∈ M . Notice that as N 1 , . . . , N r partition N we have n = n 1 + n 2 + · · · + n r . Since i kt , j kt ∈ N k for each 1 ≤ t ≤ n k , we have that a kt θ i kt and b kt θ j kt are elements of M m k . We may thus suppose further that for via σ k ∈ Ψ m k ,m k , say (where if a k is a 0-tuple, then we take σ k to be the identity of N m k ). For each 1 ≤ k ≤ r and each i ∈ N k , let Hence, by Condition (3.4) on Ψ, there exists an automorphism φ of M extending each φ i . For any 1 ≤ k ≤ r and any 1 ≤ t ≤ n k we have and so a ∼ M,n b via φ. SinceN is ℵ 0 -categorical and each M i are ℵ 0 -categorical over Ψ i,i , the conditions imposed on the tuples a and b have finitely many choices, and so Notice that, by Corollary 2.9, the structure N in the lemma above can simply be a set. In most cases we take M = M , and the result simplifies accordingly by the RNT as follows.
Example 3.6. Corollary 3.5 could be used to prove more efficiently how ℵ 0 -categoricity interplays with the greatest 0-direct decomposition of a semigroup with zero [5, Proposition 5.6]. Indeed, if S = 0 i∈I S i is the greatest 0-direct decomposition of S, and I 1 , . . . , I n is a finite partition of I corresponding to the isomorphism types of the summands of S, then it is a simple exercise to show that S = {S i : i ∈ I} is an (S; (I; I 1 , . . . , I n ); Ψ)-system, where Ψ is the collection of all isomorphisms between summands. Since (I;

Strong semilattices of semigroups
In this section we study the ℵ 0 -categoricity of strong semilattices of semigroups by making use of our most recent methodology. We are motivated by the work of the author in [18] and [19], where the homogeneity of bands and inverse semigroups are shown to depend heavily on the homogeneity of strong semilattices of rectangular bands and groups, respectively. Recall that a structure is homogeneous if every isomorphism between finitely generated substructures extend to an automorphism. A uniformally locally finite homogeneous structure is ℵ 0 -categorical [15, Corollary 3.1.3]. Consequently, each homogeneous band is ℵ 0 -categorical, although the same is not true for homogeneous inverse semigroups.
Let Y be a semilattice. To each α ∈ Y associate a semigroup S α , and assume that On the set S = α∈Y S α define a multiplication by for a ∈ S α , b ∈ S β , and denote the resulting structure by S = [Y ; S α ; ψ α,β ]. Then S is a semigroup, and is called a strong semilattice Y of the semigroups S α (α ∈ Y ). The semigroups S α are called the components of S. We follow the convention of denoting an element a of S α as a α . The We build automorphisms of strong semilattices of semigroups in a natural way using the following well known result. A proof can be found in [17].
be a strong semilattices of semigroups. Let π ∈ Aut(Y ) and, for each α ∈ Y , let θ α : S α → S απ be an isomorphism. Assume further that for any α ≥ β, the diagram Unfortunately, not all automorphisms of strong semilattices of semigroups can be constructed as in Theorem 4.1. We shall call a strong semilattice of semigroups S automorphism-pure if every automorphism of S can be constructed as in Theorem 4.1. For example, every strong semilattice of completely simple semigroups is automorphismpure [16, Lemma IV.1.8], and so both strong semilattices of groups (Clifford semigroups) and strong semilattices of rectangular bands (normal bands) are automorphism-pure.
Let S = [Y ; S α ; ψ α,β ] be a strong semilattice of semigroups. We denote the equivalence relation on Y corresponding to isomorphism types of the semigroups S α by η S , so that α η S β ⇔ S α ∼ = S β . We let Y S denote the set extension of Y given by Y S = (Y ; Y /η S ).
Hence S α θ = S β , and the claim follows. Consequently, by the ℵ 0 -categoricity of S and Proposition 2.
A natural question arises: how can we built an ℵ 0 -categorical strong semilattice of semigroups from an ℵ 0 -categorical semilattice and a collection of ℵ 0 -categorical semigroups? In this paper we will only be concerned with the ℵ 0 -categoricity of strong semilattices of semigroups in which all connecting morphisms are injective or all are constant. For arbitrary connecting morphisms, the problem of assessing ℵ 0 -categoricity appears to be difficult to capture in a reasonable way. Examples of more complex ℵ 0 -categorical strong semilattices of semigroups arise from [18], where the universal normal band is shown to have surjective but not injective connecting morphisms. We first study the case where each connecting morphism is a constant map.
Suppose that Y is a semilattice and, for each α ∈ Y , S α is a semigroup containing an idempotent e α . For each α ∈ Y let ψ α,α be the identity automorphism of S α , and for α > β let ψ α,β be the constant map with image {e β }. We follow the notation of [23] and let ψ α,β := C α,e β for each α > β in Y . It is easy to check that ψ α,β ψ β,γ = ψ α,γ for all forms a strong semilattice of semigroups. We call S a constant strong semilattice of semigroups.
is a constant strong semilattice of semigroups, then we denote the subset of Iso(S α ; S β ) consisting of those isomorphisms which map e α to e β as Iso(S α ; S β ) [eα;e β ] . Notice that the set Iso(S α ; S α ) [eα;eα] is simply the subgroup Aut(S α ; {e α }) of Aut(S α ). We may then define a relation υ S on Y by The relation υ S is reflexive since 1 Sα ∈ Aut(S α ; {e α }) for each α ∈ Y , and it easily follows that υ S forms an equivalence relation on Y .
Example 4.9. We use the work of Apps [1] to construct examples of ℵ 0 -categorical Eunitary Clifford semigroups as follows. Let G be an ℵ 0 -categorical group and H 1 < H 2 < · · · a characteristic series in G, so that each H i is a characteristic subgroup of G and H i is a subgroup of H i+1 . Apps proved that such a series must be finite, and there exists a characteristic series {1} = G 0 < G 1 < G 2 < · · · < G n = G with each G i /G i−1 a characteristically simple ℵ 0 -categorical group. For each 0 ≤ i ≤ n, let K i = G i × {i} be an isomorphic copy of G i . For each 0 ≤ i ≤ j ≤ n, let ψ i,j : K i → K j be the map given by (x, i)ψ i,j = (x, j). Then we may form a strong semilattice of the groups {0, 1, . . . , n} with the reverse ordering 0 > 1 > 2 > · · · > n. Notice that S is E-unitary as each connecting morphism is injective. If S = [Y ; S α ; ψ α,β ] is such that each connecting morphism is an isomorphism, then Y /ξ S = {Y }, and so the result above simplifies accordingly. However we can prove a more general result directly (without the condition that Y has a zero) with aid of the following proposition. The result is folklore, but a proof can be found in [17]. Proposition 4.10. Let S = [Y ; S α ; ψ α,β ] be such that each ψ α,β is an isomorphism. Then S ∼ = S α × Y for any α ∈ Y . Conversely, if T is a semigroup and Z is a semilattice then T × Z is isomorphic to a strong semilattice of semigroups such that each connecting morphism is an isomorphism.
Proof. By Proposition 4.10, S is isomorphic to S α × Y for any α ∈ Y . The first half of the result then follows as ℵ 0 -categoricity is preserved by finite direct product [8].
If S is automorphism-pure then the converse holds by Proposition 4.2, as (Y ; Y /η S ) being ℵ 0 -categorical clearly implies Y is ℵ 0 -categorical.

ℵ 0 -categorical Rees matrix semigroups
A semigroup S is called simple (0-simple) if it has no proper ideals (if its only proper ideal is {0} and S 2 = {0}). A simple (0-simple) semigroup is called completely simple (completely 0-simple) if contains a primitive idempotent, i.e. a non-zero idempotent e such that for any non-zero idempotent f of S, [20], to study the ℵ 0 -categoricity of a completely 0-simple semigroup, it is sufficient to consider Rees matrix semigroups:

By Rees Theorem
Theorem 5.1 (The Rees Theorem). Let G be a group, let I and Λ be non-empty index sets and let P = (p λ,i ) be an Λ × I matrix with entries in G ∪ {0}. Suppose no row or column of P consists entirely of zeros (that is, P is regular). Let S = (I × G × Λ) ∪ {0}, and define multiplication * on S by Then S is a completely 0-simple semigroup, denoted M 0 [G; I, Λ; P ], and is called a (regular) Rees matrix semigroup (over G). Conversely, every completely 0-simple semigroup is isomorphic to a Rees matrix semigroup.
The matrix P is called the sandwich matrix of S. Note that if S is a completely simple semigroup, then S 0 is isomorphic to a Rees matrix semigroup with sandwich matrix without zero entries [13,Section 3.3]. Consequently, by the Rees Theorem and [5, Corollary 2.12], to examine the ℵ 0 -categoricity of both completely simple and completely 0-simple semigroups, it suffices to study Rees matrix semigroups (having 0, as in our convention above).
A fundamental discovery in [5] was that to understand the ℵ 0 -categoricity of an arbitrary semigroup, it is necessary to study ℵ 0 -categorical completely (0-)simple semigroups. Indeed, they arise as principal factors of an ℵ 0 -categorical semigroup, as well as giving examples of 0-direct indecomposable summands in a semigroup with zero.
In [5] the ℵ 0 -categoricity of Rees matrix semigroups over identity matrices (known as Brandt semigroups) were determined, although we deferred the general case to this current article.
Given a Rees matrix semigroup S = M 0 [G; I, Λ; P ] with P = (p λ,i ), we let G(P ) denote the subset of G of all non-zero entries of P , that is, G(P ) := {p λ,i : p λ,i = 0}. The idempotents of S are easily described [13, Page 71]: Since there exists a simple isomorphism theorem for Rees matrix semigroups [13, Theorem 3.4.1] (see Theorem 5.9), we should be hopeful of achieving a thorough understanding of ℵ 0 -categorical Rees matrix semigroups via the RNT. However, from the isomorphism theorem it is not clear how the ℵ 0 -categoricity of the semigroup M 0 [G; I, Λ; P ] effects the sets I and Λ. We instead follow a technique of Graham [6] and Houghton [11] of constructing a bipartite graph from the sets I and Λ.
A bipartite graph is a (simple) graph whose vertices can be split into two disjoint nonempty sets L and R such that every edge connects a vertex in L to a vertex in R. The sets L and R are called the left set and the right set, respectively. Formally, a bipartite graph is a triple Γ = L, R, E such that L and R are non-empty trivially intersecting sets and E ⊆ {{x, y} : x ∈ L, y ∈ R}.
We call L∪R the set of vertices of Γ and E the set of edges. An isomorphism between a pair of bipartite graphs Γ = L, R, E and Γ = L , R , E is a bijection ψ : L ∪ R → L ∪ R such that Lψ = L , Rψ = R , and {l, r} ∈ E if and only if {lψ, rψ} ∈ E . We are therefore regarding bipartite graphs in the signature L BG = {Q L , Q R , E}, where Q L and Q R are unary relations, which correspond to the sets L and R, respectively, and E is a binary relation corresponding to the edge relation (here we abuse the notation somewhat by letting E denote the edge relation and the set of edges).
Let Γ = L, R, E be a bipartite graph. Then Γ is called complete if, for all x ∈ L, y ∈ R, we have {x, y} ∈ E. If E = ∅ then Γ is called empty. If each vertex of Γ is incident to exactly one edge, then Γ is called a perfect matching. The complement of Γ is the bipartite graph L, R, E with Hence an empty bipartite graph is the complement of a complete bipartite graph, and viceversa. We call Γ random if, for each k, ∈ N, and for every distinct x 1 , . . . , x k , y 1 , . . . , y in L (in R) there exists infinitely many u ∈ R (u ∈ L) such that {u, x i } ∈ E but {u, y j } ∈ E for each 1 ≤ i ≤ k and 1 ≤ j ≤ .
Clearly, for each pair n, m ∈ N * = N ∪ {ℵ 0 }, there exists a unique (up to isomorphism) complete biparite graph with left set of size n and right set of size m, which we denote as K n,m . There also exists a unique, up to isomorphism, perfect matching with left and right sets of size n, denoted P n . Similar uniqueness holds for the empty bipartite graph E n,m with left set of size n and right set of size m, and the complement of the perfect matching P n , which we denote as CP n . Less obviously, any pair of random bipartite graphs are isomorphic [3].

Theorem 5.2. [4]
A bipartite graph is homogeneous if and only if it is isomorphic to either K n,m , E n,m , P n , CP n for some n, m ∈ N * , or the random bipartite graph.
Since bipartite graphs are relational structures with finitely many relations, homogeneous bipartite graphs are uniformally locally finite, and thus ℵ 0 -categorical [ For any automorphism φ of Γ and x, y ∈ Γ we have that (x, v 2 , . . . , v n−1 , y) is a path in Γ if and only if (xφ, v 2 φ, . . . , v n−1 φ, yφ) is a path in Γ, since φ preserves edges and non-edges. Hence x 1 y if and only if xφ 1 yφ, and so there exists a bijection π of A such that Γ i φ = Γ iπ for each i ∈ I. We have thus proven the reverse direction of the following result, the forward being immediate. Proof. (⇒) By Proposition 5.3 we have that, for any choice of x i ∈ Γ i (i ∈ A), the set {(Γ i , x i ) : i ∈ A} forms a system of 1-pivoted p.r.c. sub-bipartite graphs of Γ. The result then follows from Proposition 2.3.
(⇐) First we show that C(Γ) forms a (Γ;Ā; Ψ)-system in Γ for someĀ and Ψ. Let A 1 , . . . , A r be the finite partition of A corresponding to the isomorphism types of the connected components of Γ, that is, Γ i ∼ = Γ j if and only if i, j ∈ A k for some k. Fix A = (A; A 1 , . . . , A r ). For each i, j ∈ A, let Ψ i,j = Iso(Γ i ; Γ j ) and fix Ψ = i,j∈A Ψ i,j . Then Ψ clearly satisfy Conditions (A), (B) and (C). Let π ∈ Aut(Ā) and, for each i ∈ A, let φ i ∈ Ψ i,iπ . Then by Proposition 5.3, φ = i∈A φ i is an automorphism of Γ, and so Ψ satisfies Condition (D). Hence C(Γ) forms an (Γ;Ā; Ψ)-system. Each Γ i is ℵ 0 -categorical (over Ψ i,i = Aut(Γ i )) andĀ is ℵ 0 -categorical by Corollary 2.9, and so Γ is ℵ 0 -categorical by Corollary 3.5. The above construct has long been fundamental to the study of Rees matrix semigroups, and has its roots in a paper by Graham in [6]. Here, it is used to describe the maximal nilpotent subsemigroups of a Rees matrix semigroup, where a semigroup is nilpotent if some power is equal to {0}. All maximal subsemigroups of a finite Rees matrix semigroup were described in the same paper, a result which was later extended in [7] to arbitrary finite semigroups. In [12], Howie used the induced bipartite graph to describe the subsemigroup of a Rees matrix semigroup generated by its idempotents. Finally, in [11], Houghton described the homology of the induced bipartite graph, and a detailed overview of his work is given in [21].   1 , g 1 , λ 1 ), . . . , (i n , g n , λ n )) of S * , we denote Γ(a) as the 2n-tuple (i 1 , λ 1 , . . . , i n , λ n ) of Γ(P ).
The composition and inverses of isomorphisms between Rees matrix semigroups behave in a natural way as follows, and a proof can be found in [17].
Since each entry of an n-tuple of Γ lies in either L or R we have that |Γ n /σ Γ,n | = 2 n , for each n. Moreover, as the automorphisms of Γ fixes the sets L and R, it easily follows that ∼ Γ,n ⊆ σ Γ,n .
In the next subsection we construct a counterexample to the converse of Proposition 5.11. Our method will be to transfer the concept of the connected components of bipartite graphs to corresponding subsemigroups of Rees matrix semigroups.
That is, P is the block matrix We denote S by G k∈A S k . The subsemigroups S k of S are called Rees components of S. Notice that each Γ(P k ) is a union of connected components of Γ(P ). The subsemigroup S k will be called a connected Rees component of S if Γ(P k ) is connected (and is therefore a connected component of Γ(P )).
Conversely, for any Rees matrix semigroup S = M 0 [G; I, Λ; P ] there exists partitions {I k : k ∈ A} and {Λ k : k ∈ A} of I and Λ, respectively, such that C(Γ(P )) = {Λ k ∪ I k : k ∈ A}. Consequently, for each k ∈ A, the subsemigroup S k = M 0 [G; I k , Λ k ; P k ] of S is a connected Rees component, where P k is the Λ k × I k submatrix of P , and are such that S k S = 0 for all k = . Following the work of Graham [6], we may then permute the rows and columns of P if necessary to assume w.l.o.g. that P is a block matrix of the form (5.1).
Note that if S is a Rees matrix semigroup with connected Rees components {S k : k ∈ A} then clearly Using the fact that automorphisms of Γ(P ) arise as collections of isomorphisms between its connected components, we obtain an alternative description of automorphisms of a Rees matrix semigroups. The proof is a simple exercise, and can be found in [17].
Corollary 5.12. Let S = G k∈A S k = M 0 [G; I, Λ; P ] be a Rees matrix semigroup such that each S k = M 0 [G; I k , Λ k ; P k ] is a connected Rees component of S. Let π be a bijection of A and, for each k ∈ A, let φ k = (θ, ψ k , (u λ . Moreover, every automorphism of S can be described in this way.
We observe that the induced group automorphisms of the isomorphisms φ k above must all be equal.
Recall that if S = M 0 [G; I, Λ; P ] is ℵ 0 -categorical, then Γ(P ) is ℵ 0 -categorical by Proposition 5.11, and thus C(Γ(P )) is finite, up to isomorphism, with each connected component being ℵ 0 -categorical by Proposition 5.4. We extend this result to the set of all connected Rees components of S as follows: Proposition 5.13. Let S = G k∈A S k be an ℵ 0 -categorical Rees matrix semigroup such that each S k is a connected Rees component of S. Then each S k is ℵ 0 -categorical and S has finitely many connected Rees components, up to isomorphism.
Proof. We claim that {(S k , a k ) : k ∈ A} is a system of 1-pivoted p.r.c. subsemigroups of S for any a k ∈ S * k , to which the result follows by Proposition 2.3. Indeed, let φ be an automorphism of S such that a k φ = a l for some k, l. Then, by Corollary 5.12, there exists a bijection π of A with S k φ = S kπ = S l as required.
Our interest is now in attaining a converse to the proposition above, since it would provide us with a method for building 'new' ℵ 0 -categorical Rees matrix semigroups from 'old'. With the aid of Lemma 3.4, we shall prove that a converse exists in the class of Rees matrix semigroups over finite groups. The case where the maximal subgroups are infinite is an open problem.
We are now able to prove our desired converse to Proposition 5.13 in the case where the maximal subgroups are finite. (⇐) Since S is regular with finite maximal subgroups, to prove S is ℵ 0 -categorical, it suffices by [5,Corollary 3.13] to show that |E(S) n / ∼ S,n | is finite, for each n ∈ N. Let {S k : k ∈ A} be the set of connected Rees components of S, which is finite up to isomorphism and with each S k being ℵ 0 -categorical. Define a relation η on A by i η j if and only if Iso(S i ; S j )(1 G ) = ∅. By Corollary 5.10 we have that η is an equivalence relation.
We first prove that A/η is finite. Suppose for contradiction that there exists an infinite set X of pairwise η-inequivalent elements of A. Since S has finitely many connected components up to isomorphism, there exists an infinite subset {i r : r ∈ N} of X such that S in ∼ = S im for each n, m. Fix an isomorphism φ in : S in → S i 1 for each n ∈ N. Then as Aut(G) is finite there exists distinct n, m such that φ G in = φ G im , and so φ in φ −1 im ∈ Iso(S in ; S jm )(1 G ) by Corollary 5.10. Hence i n η i m , a contradiction, and so A/η is finite.
Let S = k∈A S k , noting that S is the 0-direct union of the S k , and in particular is a subsemigroup of S. Let A/η = {A 1 , . . . , A r } and setĀ = (A; A 1 , . . . , A r ). For each i, j ∈ A, let Ψ i,j = Iso(S i ; S j )(1 G ) and fix Ψ = i,j∈A Ψ i,j . We prove that {S k : k ∈ A} forms an (S; S ;Ā; Ψ)-system in S. First, by our construction, if i, j ∈ A m for some m then Ψ i,j = ∅, and so Ψ satisfies Condition (3.1). Furthermore, it follows immediately from Corollary 5.10 that Ψ satisfies Conditions (3.2) and (3.3). Finally, take any π ∈ Aut(Ā) and, for each k ∈ A, let φ k ∈ Ψ k,kπ . Then as φ G k = 1 G for each k ∈ A, we may construct an automorphism φ of S from the set of isomorphisms {φ k : k ∈ A} by Corollary 5.12. Hence, as φ extends each φ k by construction, we have that {S k : k ∈ A} forms an (S; S ;Ā; Ψ)system as required. Since S k is ℵ 0 -categorical, it is ℵ 0 -categorical over Ψ k,k = Aut(S k )(1 G ) by Lemma 5.14. By Corollary 2.9Ā is ℵ 0 -categorical, and so |(S ) n / ∼ S,n | < ℵ 0 by Lemma 3.4. Given that E(S) ⊆ S by (5.2), we therefore have that Hence S is ℵ 0 -categorical.
Open Problem 5.16. Does Theorem 5.15 hold if G is allowed to be any ℵ 0 -categorical group?
We now have a simple tool for constructing a counterexample to the converse of Proposition 5.11. Indeed, by Proposition 5.13, it suffices to find a Rees matrix semigroup over an ℵ 0 -categorical group with ℵ 0 -categorical induced bipartite graph, but with infinitely many non-isomorphic connected Rees components.
Example 5.17. Let G be an ℵ 0 -categorical infinite abelian group with identity element 1, and {g i : i ∈ N} be an enumeration of its non-identity elements (such a group exists by the work of Rosenstein [22], taking G = N Z 2 , for example). Let I k = {i k s : s ∈ N} and Λ k = {λ k t : t ∈ N} be infinite sets for each k ∈ N.
Then each Γ(P k ) is a complete bipartite graph, isomorphic to K ℵ 0 ,ℵ 0 , and is thus ℵ 0categorical by Theorem 5.2. For each k ∈ N, let S k be the connected Rees matrix semigroup [G; I k , Λ k ; P k ], and set G k∈N S k = M 0 [G; I, Λ; P ]. Then Γ(P ), being the disjoint union of the pairwise isomorphic ℵ 0 -categorical bipartite graphs Γ(P k ), is ℵ 0 -categorical by Theorem 5.4.
We claim that S k ∼ = S if and only if k = l. Let (θ, ψ, (u i ) i∈I k , (v λ ) λ∈Λ k ) be an isomorphism between S k and S , and assume w.l.o.g. that k ≥ . Since there exists only finitely many rows of P k and P which have non-identity entries, there exists λ k s ∈ Λ k such that both row λ k s of P k and row λ k s ψ of P consist entirely of identity entries. Hence, for each i k t ∈ I k , p u, say. Dually, by considering the columns of P k and P , we have We therefore have, for each 1 ≤ m ≤ k, . . , g } as G is abelian. It follows that the automorphism θ maps {g 1 , . . . , g k } to {g 1 , . . . , g }. Since k ≥ , this forces k = l, thus proving the claim. We have shown that M 0 [G; I, Λ; P ] has infinitely many non-isomorphic connected Rees components, and is therefore not ℵ 0categorical by Proposition 5.13.

5.2.
Labelled bipartite graphs. In Example 5.17, the problem which arose was that by shifting from the sandwich matrix P = (p λ,i ) to the induced bipartite graph Γ(P ) we have "forgotten" the value of the entries p λ,i . In this subsection we extend the construction of the induced bipartite graph of a Rees matrix semigroup to attempt to rectifying this problem, as well as to build classes of ℵ 0 -categorical Rees matrix semigroups. Further examples of ℵ 0 -categorical Rees matrix semigroups can then be built using Theorem 5.15. This gives rise to a natural signature in which to consider Σ-labelled bipartite graphs as follows. For each σ ∈ Σ, take a binary relation symbol E σ and let Then we call L BGΣ the signature of Σ-labelled bipartite graphs, where (x, y) ∈ E σ if and only if {x, y} ∈ E and {x, y}f = σ.
Let Γ f be a Σ-labelled bipartite graph. Then for any set Σ and bijection g : Σ → Σ , we can form a Σ -labelling of Γ simply by taking Γ f g , which we call a relabelling of Γ f . Notice that if ψ is an automorphism of Γ, then ψ ∈ Aut(Γ f ) if and only if ψ ∈ Aut(Γ f g ). Indeed, if ψ ∈ Aut(Γ f ) then for any edge {x, y} of Γ we have since g is a bijection. The converse is proven similarly, and the following result is then immediate from the RNT. Lemma 5.19. Let Γ f be a Σ-labelling of a bipartite graph Γ. Then Γ f is ℵ 0 -categorical if and only if any relabelling of Γ f is ℵ 0 -categorical. Lemma 5.20. If Γ f = (Γ, Σ, f ) is an ℵ 0 -categorical labelled bipartite graph then Σ is finite and Γ is ℵ 0 -categorical.
Proof. For each σ ∈ Σ, let {x σ , y σ } be an edge in Γ such that {x σ , y σ }f = σ. Then {(x σ , y σ ) : σ ∈ Σ} is a set of distinct 2-automorphism types of Γ f , and so Σ is finite by the RNT. Since automorphisms of Γ f induce automorphisms of Γ, the final result is immediate from the RNT.
A consequence of the previous pair of lemmas is that, in the context of ℵ 0 -categoricity, it suffices to consider finitely labelled bipartite graphs, with labelling set m = {1, 2, . . . , m} for some m ∈ N.
Lemma 5. 21. Let Γ f = ( L, R, E , m, f ) be an m-labelled bipartite graph such that either L or R are finite. Then Γ f is ℵ 0 -categorical.
Proof. Without loss of generality assume that L = {l 1 , l 2 , . . . , l r } is finite. Define a relation τ on R by y τ y if and only if y and y are adjacent to the same elements in L and {l i , y}f = {l i , y }f for each such l i ∈ L. Note that since both L and m are finite, R has finitely many τ -classes, say R 1 , . . . , R t . Considering R simply as a set, fix A = (R; R 1 , . . . , R t ).
Since L is finite, to prove that Γ f is ℵ 0 -categorical it suffices to show that (Γ f \ L) n = R n has finitely many ∼ Γ f ,n -classes for each n ∈ N by a simple generalization of [5,Proposition 2.11]. Let a = (r 1 , . . . , r n ) and b = (r 1 , . . . , r n ) be n-tuples of R such that a ∼ A,n b via ψ ∈ Aut(A), say. We claim that the mapψ : Γ f → Γ f which fixes L and is such that ψ| R = ψ is an automorphism of Γ f . Indeed, as ψ setwise fixes the τ -classes, we have (λ, λψ) ∈ τ for each λ ∈ R. Hence λ and λψ are adjacent to the same elements in L, and so λψ}f , so thatψ preserves labels. This proves the claim.
For each 1 ≤ k ≤ n we have r kψ = r k ψ = r k , so that a ∼ Γ f ,n b. Consequently, The set extension A is ℵ 0 -categorical by Corollary 2.9, and so |A n / ∼ A,n | is finite for each n ≥ 1. Hence Γ f is ℵ 0 -categorical.
Lemma 5. 22. Let Γ f = ( L, R, E , m, f ) be such that there exists p ∈ m with {x, y}f = p for all but finitely many edges in Γ. Then Γ f is ℵ 0 -categorical if and only if Γ is ℵ 0categorical.
Proof. Suppose Γ is ℵ 0 -categorical, and that {l 1 , r 1 }, . . . , {l t , r t } are precisely the edges of Γ such that {l k , r k }f = p, where l k ∈ L and r k ∈ R. Let a and b be n-tuples of Γ f such that (a, l 1 , r 1 , . . . , l t , r t ) ∼ Γ,n+2t (b, l 1 , r 1 , . . . , l t , r t ) via ψ ∈ Aut(Γ), say. We claim that ψ is an automorphism of Γ f . For each 1 ≤ k ≤ t we have l k ψ = l k and r k ψ = r k so that {l k , r k }f = {l k ψ, r k ψ}f.
It follows that {l, r}f = p if and only if {lψ, rψ}f = p, and so ψ preserves all labels, thus proving the claim. Consequently, a ∼ Γ f ,n b via ψ, so that The converse is immediate from Lemma 5.20.
Definition 5.23. Given a Rees matrix semigroup S = M 0 [G; I, Λ; P ], we form a G(P )labelling of the induced bipartite graph Γ(P ) = I, Λ, E of S in the natural way by taking the labelling f : E → G(P ) given by We denote the labelled bipartite graph by Γ(P ) l , which we call the induced labelled bipartite graph of S.
The converse however fails to hold in general, and a counterexample will be constructed at the end of the section. Despite this, the proposition above enables us to produce concrete examples of ℵ 0 -categorical Rees matrix semigroups. For example, the result below is immediate from Lemma 5.21.
Corollary 5.25. Let S be a Rees matrix semigroup over an ℵ 0 -categorical group having sandwich matrix P with finitely many rows or columns, and G(P ) being finite. Then S is ℵ 0 -categorical.
Similarly, Lemma 5.22 may be used in conjunction with Proposition 5.24 to obtain: Corollary 5.26. Let S = M 0 [G; I, Λ; P ] be a Rees matrix semigroup such that G and Γ(P ) are ℵ 0 -categorical, and all but finitely many of the non-zero entries of P are the identity of G. Then S is ℵ 0 -categorical.
Following [14], we call a completely 0-simple semigroup S pure if it is isomorphic to a Rees matrix semigroup with sandwich matrix over {0, 1}. In [11], Houghton considered trivial cohomology classes of Rees matrix semigroups, a property which is proven in Section 2 of his article to be equivalent to being pure. Hence, by [ It follows that all orthodox completely 0-simple semigroups are necessarily pure, but the converse is not true in general. Indeed, a completely 0-simple semigroup is orthodox if and only if it is isomorphic to a Rees matrix semigroup with sandwich matrix over {0, 1} and with induced bipartite graph a disjoint union of complete bipartite graphs [9,Theorem 6]. Hence, in this case, it can be easily shown that the isomorphism types of the connected Rees components depends only on the isomorphism types of the induced (complete) bipartite graphs.
We observe that if the sandwich matrix of a Rees matrix semigroup is over {0, 1} then Γ(P ) l is simply labelled by {1}. Therefore all automorphisms of Γ(P ) automatically preserve the labelling, and so Γ(P ) l is ℵ 0 -categorical if and only if Γ(P ) is ℵ 0 -categorical. The following result is then immediate from Proposition 5.11 and Corollary 5.26. Furthermore, since complete bipartite graphs are ℵ 0 -categorical by Theorem 5.2, a disjoint union of complete bipartite graphs is ℵ 0 -categorical if and only if it has finitely many connected components, up to isomorphism, by Proposition 5.4. The corollary above thus reduces in the orthodox case as follows.
Corollary 5.28. Let S = M 0 [G; I, Λ; P ] be an orthodox Rees matrix semigroup. Then S is ℵ 0 -categorical if and only if G is ℵ 0 -categorical and Γ(P ) has finitely many connected components, up to isomorphism.
In [5] we studied inverse completely 0-simple semigroups, that is, Brandt semigroups. These are necessarily orthodox, and are isomorphic to a Rees matrix semigroup of the form M 0 [G; I, I; P ] where P is the identity matrix, that is, p ii = 1 and p ij = 0 for each i = j in I, and are denoted B 0 [G; I]. Since the induced biparite graph of a Brandt semigroup is a perfect matching, it is ℵ 0 -categorical by Theorem 5.2. Corollary 5.28 then simplifies to obtain our classification of ℵ 0 -categorical Brandt semigroups [5,Proposition 4.3], which states that a Brandt semigroup over a group G is ℵ 0 -categorical if and only if G is ℵ 0categorical.
We are now able to construct a simple counterexample to the converse of Proposition 5.24. Let G = {g i : i ∈ N} be an infinite ℵ 0 -categorical group. Let where Q = (q i,j ) is such that q i,i = g i and q i,j = 0 for each i = j. Then Γ(P ) = Γ(Q) (and are isomorphic to P N ) and (1 G , 1 Γ(P ) , (g −1 i ) i∈N , (1) λ∈N ) is an isomorphism from S to T by Theorem 5.9 since p i,i 1 G = 1 = g i g −1 i = 1 · q i,i · g −1 i , for each i ∈ N. Since S is ℵ 0 -categorical by the ℵ 0 -categoricity of G, the same is true of T . However, Γ(Q) l is a G-labelling, and is thus not ℵ 0 -categorical by Lemma 5.20. Hence T is our desired counterexample.
Open Problem 5.29. Does there exist an ℵ 0 -categorical connected Rees matrix semigroup over a finite group which is not isomorphic to a Rees matrix semigroup with ℵ 0 -categorical induced labelled bipartite graph?
To further incorporate the link between the induced bipartite graph of a Rees matrix semigroup and the entries of the sandwich matrix, we could instead introduce the stronger notion of an induced group labelled bipartite graph. A group labelled bipartite graph is a Glabelled bipartite graph Γ f = ( L, R, E , G, f ), for some group G, where an automorphism of Γ f is a pair (ψ, θ) ∈ Aut(Γ) × Aut(G) such that, for each ∈ L, r ∈ R, ( , r)f = g ⇔ ( ψ, rψ)f = gθ.
However, group labelled biparite graphs do not appear to be first order structures.
Let S = M 0 [G; I, Λ; P ] be such that G(P ) forms a subgroup of G. Then we may define the induced group labelled bipartite graph of S as the G(P )-labelled bipartite graph Γ(P ) f , with automorphisms being pairs (ψ, θ) ∈ Aut(Γ) × Aut(G(P )) such that p λψ,iψ = p λ,i θ for each i ∈ I, λ ∈ Λ. Notice that if (ψ, θ) is an automorphism of the induced group labelled bipartite graph of S and is such that θ extends to an automorphism θ of G, then (θ , ψ, (1) i∈I , (1) λ∈Λ ) is clearly an automorphism of S. However, we do not in general obtain all automorphisms of S in this way. Similar problems therefore arise in regard to when ℵ 0 -categoricity of S passes to its induced group labelled bipartite graph (by which we mean the induced group labelled bipartite graph has an oligomorphic automorphism group).