Laws for generalised interior operations in semigroups

Abstract

Motivated by interior operators and by analogous generalisations of closure operators to semigroups in general, we define left, right and two-sided interior operations on a semigroup S in terms of a distinguished set of idempotents E using the natural order. In the left-sided case, we require that for all $s\in S$, there is a largest $e\in E$ under the natural order such that $es=e$. We generalise them using suitable Green-like relations, and characterise the resulting classes of unary and biunary semigroups as quasivarieties or varieties.

1 Preliminaries on sets of idempotent elements

First, we introduce the (well-known) partial orders and quasiorders on the set of idempotents E(S) of the semigroup S that we will be using throughout.

• The natural right quasiorder is given by $e\le _r f$ if and only if $e=ef$;

• the natural left quasiorder is given by $e\le _l f$ if and only if $e=fe$;

• $\sim _r$ and $\sim _l$ denote the respective induced equivalence relations;

• the natural order is given by denote by $e\le f$ if and only if $e=ef=fe$ (the intersection of $\le _r$ and $\le _l$), and is a partial order.

If $E\subseteq E(S)$, then E is said to be

• right reduced if ${\le _r}\subseteq {\le _l}$ (that is, $e=ef$ implies $e=fe$ for all $e,f\in E$);

• left reduced if ${\le _l}\subseteq {\le _r}$;

• reduced if the two quasiorders are equal;

• right pre-reduced if $\le _r$ is partial order on E (that is, $e\sim _r f$ implies $e=f$ for all $e,f\in E$, or equivalently, $e=ef$ and $f=fe$ imply $e=f$ for all $e,f\in E$); left pre-reduced if $\le _l$ is partial order on E;

• pre-reduced it is both right and left pre-reduced;

• maximal right pre-reduced if E is right pre-reduced and is not contained in a strictly larger right pre-reduced set of idempotents.

Obviously, if E is (left/right) reduced, then it is (left/right) pre-reduced (although the converses fail). A right pre-reduced subset E of E(S) is nothing other than a selection of at most one idempotent from each $\sim _r$-class in E(S); hence a typical maximal right pre-reduced subset E of E(S) may be obtained by selecting precisely one idempotent from each $\sim _r$-class of E(S), as is easily seen. Note that if $E=E(S)$ is right pre-reduced (resp. left pre-reduced, pre-reduced), then it is automatically right reduced (resp. left reduced, reduced), as is well-known and easily shown.

2 Motivation

The concept of a closure operator is a familiar one: it is an operator given by $A\mapsto {{\,\textrm{cl}\,}}(A)$ on the power set $2^X$ of the set X, satisfying the following familiar laws (expressed in terms of intersection):

$$\begin{aligned}A\cap {{\,\textrm{cl}\,}}(A)=A,\ {{\,\textrm{cl}\,}}(A\cap B)\cap {{\,\textrm{cl}\,}}(B)={{\,\textrm{cl}\,}}(A\cap B),\ {{\,\textrm{cl}\,}}({{\,\textrm{cl}\,}}(A))={{\,\textrm{cl}\,}}(A).\end{aligned}$$

Defining the “closed sets" in X to be those of the form ${{\,\textrm{cl}\,}}(A)$, $A\subseteq X$, it follows easily that for any $A\in 2^X$, ${{\,\textrm{cl}\,}}(A)\in 2^X$ is the smallest closed subset of X containing A. Conversely, one can show that every closure operator on the set X arises in this way from a collection of “closed subsets" of X such that every subset of X is contained in a smallest closed subset. The case of topological closure operators defined on general Boolean algebras has been investigated in detail; see [11], where an equational basis for such algebras was given.

An abstraction to (meet-)semilattices is straightforward, and is just a special case of the concept as it applies to partially ordered sets: C is a closure operation on the semilattice S if there is $E\subseteq S$ such that for all $s\in S$, there is a smallest $e\in E$ for which $se=s$, called C(s); then E is the collection of closed elements (of the form $C(s), s\in S$), and can be shown to be a subsemilattice of S. One can write down some simple equational laws to characterise this situation – essentially those above for closure operators but with intersections replaced by meets.

Generalising to a semigroup setting, the concept of a C-semigroup was defined in [5], although it had earlier been defined in a different way in [1]. A C-semigroup is a semigroup equipped with a kind of “closure operation", the left-sided version of which is as follows: S is a left C-semigroup if S contains a multiplicative subsemigroup $E\subseteq E(S)$ which is a subsemilattice and such that for every $a\in S$, there is a smallest $e\in E$ (under the meet-semilattice order, which is of course the natural order on E) for which $ea=a$; denote this smallest $e\in E$ by C(a). This order-theoretic approach rather than some definition in terms of laws (like those used to define closure operators) is the appropriate way to generalise the concept from semilattices to semigroups. Nevertheless, it was shown in [5] that such unary semigroups could in any case be characterised in terms of finitely many equational axioms involving the induced unary operation C and multiplication. C-semigroups were studied for their own sake in [5], with numerous examples given, such as inverse semigroups and semigroups of partial functions. Denoting by PT(X) the semigroup of partial functions acting on the right on the set X so that fg is “first f, then g" (here and throughout), then letting E be the semilattice under composition of all restrictions of the identity map on X to subsets of X, it is easy to see that C(s), the smallest member of E for which $es=s$, exists for all $s\in PT(X)$ and is just $1_{{{\,\textrm{dom}\,}}(s)}$, the identity function restricted to the domain of s in X, ${{\,\textrm{dom}\,}}(s)$. This results in the class of left restriction semigroups.

More recently, the current author became aware of semigroup settings in which the closed elements did not necessarily commute with one-another, arising naturally in other areas of mathematics. For example, consider any Rickart $*$-ring: this is an associative ring with identity and involution, such that the left (equivalently right) annihilator of every element is generated as a left (respectively right) ideal by a necessarily unique projection (symmetric idempotent e, satisfying $e^*=e$). Rickart $*$-rings arise in the theory of operator algebras. If R is one such and $s\in R$, with the projection ${{\,\textrm{ann}\,}}(s)$ generating the left annihilator $(0:s)=\{r\in R\mid rs=0\}$ of $s\in R$, then letting $D(s)=1-{{\,\textrm{ann}\,}}(s)$, we find that D(s) is the smallest $e\in E^*(R)$, the set of projections of R, under the natural order, such that $es=s$. This is formally identical to the definition of C(s) in a left C-semigroup, but note that members of $E^*(R)$ do not commute with one-another and $E^*(R)$ is not in general a multiplicative subsemigroup of R.

This leads to consideration of semigroups S, with or without involution, equipped with a subset E of idempotents for which for all $s\in S$, there is a smallest $e\in E$ under the natural order for which $es=s$. Such D-semigroups were considered in [13], where they were axiomatised as a variety of unary semigroups, and their properties studied in the semigroup and ring cases. It was noted in [13] that D-semigroups have a close connection with a class of semigroups defined by means of a generalised Green’s relation. Thus, for a semigroup S and non-empty $E\subseteq E(S)$, we define $\widetilde{{\mathcal R}}_E$ as follows: $(a,b)\in \widetilde{{\mathcal R}}_E$ if and only if for all $e\in E$, $ea=a$ if and only if $eb=b$. Then S is weakly left E-abundant (also known in the literature as “left E-semiabundant") if for all $s\in S$ there is $e\in E$ for which $(s,e)\in \widetilde{{\mathcal R}}_E$. This notion is a variant of the notion of left abundance of a semigroup, which is defined in terms of the generalised Green’s relation ${\mathcal R}^*$ given by $(a,b)\in {\mathcal R}^*$ if and only if for all $x,y\in S^1$, $xa=ya$ if and only if $xb=yb$; one says that S is left abundant if every ${\mathcal R}^*$-class contains at least one idempotent. It is easily seen that left abundance implies weak left E(S)-abundance.

Following notation introduced in Sect. 1.1 of [9], as well as in Sect. 1 of [15], we shall adopt the following conventions throughout: if S is a semigroup with $E\subseteq E(S)$ and $\rho$ an equivalence relation on S, we shall say S is

• $(E,\rho )$-abundant if every $\rho$-class of S contains an element of E;

• $\rho$-abundant if $E=E(S)$ in the above; and

• if the idempotent in each $\rho$-class in each of the above is unique we replace “abundant" with “amiable".

We shall use this standardised notation and terminology throughout what follows. In terms of this new notation, we see that S is weakly left E-abundant if and only if it is $(E,\widetilde{{\mathcal R}}_E)$-abundant, S is left abundant if and only if it is ${\mathcal R}^*$-abundant, and S is regular if and only if it is ${\mathcal R}$-abundant (equivalently, ${\mathcal L}$-abundant).

It was shown in [13] that S is a D-semigroup if and only if S is $(D(S),\widetilde{{\mathcal R}}_{D(S)})$-abundant where $D(S)=\{D(s)\mid s\in S\}$, and D(S) is left reduced. It then follows easily that for $s\in S$, D(s) is in fact the unique $e\in D(S)$ such that $(s,e)\in \widetilde{{\mathcal R}}_E$, and so S is $(D(S),\widetilde{{\mathcal R}}_{D(S)})$-amiable.

At this point a possible generalisation of D-semigroups becomes obvious: just drop the left reduced requirement but retain amiability. It is easily seen that a semigroup S is $(E,\widetilde{{\mathcal R}}_E)$-amiable if and only if S is $(E,\widetilde{{\mathcal R}}_E)$-abundant and E is left pre-reduced; in such S, define $D(s)=e$ for all $s\in S$ (where $e\in E$ is such that $(s,e)\in \widetilde{{\mathcal R}}_E$). This yields the definition of “generalised D-semigroups" given in [14]. There, it was noted that every $(E,\widetilde{{\mathcal R}}_E)$-abundant semigroup S can be made into such a generalised D-semigroup, by selecting precisely one $e\in E$ from each $\widetilde{{\mathcal R}}_E$-class at the outset to give $F\subseteq E$, and defining $D(s)=e\in F$ where $(s,e)\in \widetilde{{\mathcal R}}_E$, and moreover that $\widetilde{{\mathcal R}}_E=\widetilde{{\mathcal R}}_F$, and F is necessarily left pre-reduced. (Of course, in general there may be many ways to choose F and hence to define D.)

A generalised version of the original order-theoretic motivation can now be recovered: a generalised D-semigroup S is precisely a semigroup with distinguished set of idempotents E such that for all $s\in S$, there is a unique smallest $e\in E$ under the left quasiorder $\le _l$ on E for which $es=s$; this forces E to be left pre-reduced, $(s,D(s))\in \widetilde{{\mathcal R}}_E$ for all $s\in S$, and $D(S)=E$. The class of generalised D-semigroups has a finite equational axiomatisation as a class of unary semigroups, generalising that for C-semigroups. The class of generalised D-semigroups which are $(E,{\mathcal R}^*)$-abundant as semigroups may be characterised as a proper subquasivariety.

In the current work, our interest moves from closure-like operations on semigroups to interior-like operations. The original motivating examples come from topology and modal logic, although in those settings, closure and interior operators are essentially equivalent, since if ${{\,\text {int}\,}}(A)$ denotes the interior of the subset A of a topological space, this is nothing but the complement of ${{\,\textrm{cl}\,}}(A^c)$, where $A^c$ is the complement of A in X. More abstractly, one may define an interior algebra to be a Boolean algebra equipped with a unary operation satisfying certain laws, modelling the case of the power set algebra of a topological space equipped with its interior operator; these also arise as so-called type S4 modal Boolean algebras in modal logic. Interior algebras were studied from the varietal perspective in [2].

Generalising interior algebras to semilattices is again straightforward: we may say that the semilattice S is an interior semilattice if there is distinguished $E\subseteq E(S)$ such that for all $s\in S$, there is a largest $e\in E$ (under the meet-semilattice order on S) such that $es=e$. The passage to semigroups is then analogous to this passage from closure operations on semilattices to general C-semigroups. Thus, in [8], so-called interior semigroups were considered; an interior semigroup S is a semigroup with subsemigroup that forms a semilattice E for which, for all $s\in S$, there is a largest $e\in E$, denoted I(s), such that $es=e$; an equational axiomatisation of such interior semigroups was given in [8].

Examples of interior semigroups defined this way arise from the theory of partial functions. Denoting again by PT(X) the semigroup of partial functions on the set X, and again letting E be the semilattice under composition all restrictions of the identity map on X to subsets of X, it is easy to see that I(s), the largest member of E for which $es=e$, exists for all $s\in PT(X)$ and is just $1_{{{\,\textrm{fix}\,}}(s)}$, the identity function restricted to the fix-set of s in X, ${{\,\text {fix}\,}}(s)=\{x\in X\mid xs=x\}$.

It is natural to generalise the concept of an interior operation in the same way that C-semigroups were generalised to give D-semigroups, by dropping the requirement that E be a semilattice and instead using the natural order on E. For example, in a Rickart $*$-ring R, one can define $I(s)=1-D(1-s)={{\,\textrm{ann}\,}}(1-s)$, and then one can check that I(s) is the largest $e\in E^*(R)$ (the set of symmetric idempotents) under the natural order such that $es=e$; in terms of left annihilators, I(s) is simply the projection that generates $(0:1-s)$. This leads to consideration of any semigroup S equipped with some set of idempotents E such that for every $s\in S$, there is a largest $e\in E$ under the natural order for which $es=e$. As a special case of more general results, we give axioms for these left interior semigroups in Sect. 4 and study their basic properties.

An alternative way to generalise semilattices with a (left) interior operation (which are of course commutative) is to consider any semigroup S equipped with a set of idempotents E such that for each $s\in S$, there is an element $I(s)\in E$ which is simultaneously both the largest $e\in E$ for which $es=e$ and the largest $e\in E$ for which $se=e$. We call such semigroups two-sided interior semigroups, and also consider these in what follows.

Just as for D-semigroups, the notion of general left interior semigroup we have just defined can be described in terms of some kind of variant of a Green’s relation which will be studied in Sect. 3. The relevant analog of $\widetilde{{\mathcal R}}_E$ here is the equivalence relation ${\mathcal F}_E$, defined on any semigroup with $E\subseteq E(S)$, as follows:

$$\begin{aligned} (s,t)\in {\mathcal F}_E\hbox { if and only if, for all }e\in E, (es=e\Leftrightarrow et=e).\end{aligned}$$

There is an obvious right-sided variant ${\mathcal G}_E$, defined as follows:

$$\begin{aligned} (s,t)\in {\mathcal G}_E\hbox { if and only if, for all }e\in E, (se=e\Leftrightarrow te=e).\end{aligned}$$

The notation here stems from the use of the symbols ${\mathcal F}$ and ${\mathcal G}$ in [15] for the relations defined as follows:

$$\begin{aligned} (s,t)\in {\mathcal F}\hbox { if and only if, for all }x\in S, (xs=x\Leftrightarrow xt=x)\end{aligned}$$

and

$$\begin{aligned} (s,t)\in {\mathcal G}\hbox { if and only if, for all }x\in S, (sx=x\Leftrightarrow tx=x).\end{aligned}$$

In that work, these were compared to Green’s relations ${\mathcal L}$ and ${\mathcal R}$ as well as their variants ${\mathcal L}^*$ and ${\mathcal R}^*$, and ${\mathcal F}$-abundance and amiability of semigroups were studied, as were variants of superabundance involving ${\mathcal K}={\mathcal F}\cap {\mathcal G}$. A major difference between ${\mathcal F}$ (resp. ${\mathcal G}$) and ${\mathcal L}$ (resp. ${\mathcal R}$) is that the latter is a right (resp. left) congruence, whereas the former is not.

It follows that S is a left interior semigroup in the general sense described above if and only if S is $(I(S),{\mathcal F}_{I(S)})$-abundant (where $I(S)=\{I(s)\mid s\in S\}$) and I(S) is right reduced; again it will follow that I(s) is the unique $e\in I(S)$ such that $(s,e)\in {\mathcal F}_{I(S)}$, so that S is $(I(S),{\mathcal F}_{I(S)})$-amiable.

As for D-semigroups, an obvious generalisation beckons: in Sect. 4, we define left generalised interior semigroups (left GI-semigroups) to be nothing but $(E,{\mathcal F}_E)$-amiable semigroups (equivalently, $(E,{\mathcal F}_E)$-abundant semigroups in which E is right pre-reduced), equipped with the obvious induced unary operation I. (There is an obvious notion of right GI-semigroups, defined from $(E,{\mathcal G}_E)$-amiable semigroups in the same way.) As for generalised D-semigroups, every $(E,{\mathcal F}_E$)-abundant semigroup S can be made into a left GI-semigroup, by selecting precisely one $e\in E$ from each ${\mathcal F}_E$-class at the outset to give $F\subseteq E$, and defining $I(s)=e\in F$ where $(s,e)\in {\mathcal F}_E$; then F is necessarily right pre-reduced and ${\mathcal F}_E={\mathcal F}_F$. Again the choice of F and hence of I here is far from unique.

Again, there is a nice order-theoretic formulation of the left GI-semigroup notion. We shall show that a left GI-semigroup S is nothing but a semigroup with distinguished set of idempotents E such that for all $s\in S$, there is a unique largest $f\in E$ under the right quasiorder for which $fs=f$, namely $f=I(s)$; this forces E to be right pre-reduced, $(s,I(s))\in {\mathcal F}_E$ for all $s\in S$, and $I(S)=E$.

We shall show that the class of left GI-semigroups is a proper quasivariety, and we give a finite axiomatisation. This differentiates left GI-semigroups from generalised D-semigroups, which form a variety. We also show that the class of those left GI-semigroup S which are $(E,{\mathcal F})$-abundant is a proper subquasivariety of the class of left GI-semigroups. We shall see in Proposition 5.1 that in any left GI-semigroup S and for any $e,f\in I(S)$, I(ef) is an upper bound on all the common lower bounds of e, f; in many cases it is also a lower bound of them and hence is their meet. We give an equational description of those left GI-semigroups having such “large meets" (formally defined in Definition 5.2).

We shall consider semigroups which are both left GI-semigroups and right GI-semigroups with respect to the same set of idempotents; of particular interest are those for which the left and right interior operations are equal. We give an equational characterisation of the resulting unary semigroups, which we call two-sided GI-semigroups. In these cases, I(S) is reduced and is a meet-semilattice under the natural order having large meets in the sense of the previous paragraph. Indeed, we show that two-sided GI-semigroups are nothing but the two-sided interior semigroups discussed previously, in which for every $s\in S$, I(s) is both the largest $e\in E$ under the natural order for which $es=e$, and the largest $e\in E$ under the natural order for which $se=e$.

We conclude the paper by considering the relation ${\mathcal K}_E={\mathcal F}_E\cap {\mathcal G}_E$ for its own sake; we shall see that ${\mathcal K}_E$-abundance implies ${\mathcal K}_E$-amiability (parallelling the situation for superabundance involving ${\mathcal L}^*\cap {\mathcal R}^*$, and ${\mathcal K}$-abundance as in [15]). We give a finite properly quasiequational axiomatisation of the resulting class of unary semigroups, which contains the variety of two-sided GI-semigroups.

To conclude this section, we give two tables that summarise some of the key information. In the first of them, immediately below, we show how to define D(s) and I(s) in the main motivating examples we discussed.

$$\begin{aligned} \begin{array}{|l||c|c|} \hline \hbox {type of example}&{}D(s)&{}I(s)\\ \hline \hbox {power set of topological space}&{}{{\,\textrm{cl}\,}}(s)&{}{{\,\textrm{int}\,}}(s)\\ PT(X)&{}1_{{{\,\textrm{dom}\,}}(s)}&{}1_{{{\,\textrm{fix}\,}}(s)}\\ \hbox {Rickart }*\hbox {-ring}&{}1-{{\,\textrm{ann}\,}}(s)&{}{{\,\textrm{ann}\,}}(1-s)\\ \hline \end{array} \end{aligned}$$

In the second table, we give the three ways of defining the two operations in generalised D-semigroups and left GI-semigroups respectively: order-theoretically, via generalised Green’s relations, and via axioms. The first two ways are defined in terms of some distinguished set of idempotents E, where the order is $\le _l$ on E for D and $\le _r$ on E for I, and for the second we are assuming amiability with respect to the relevant generalised Green’s relation determined by E.

$$\begin{aligned} \begin{array}{|l||c||c|} \hline \hbox {type of definition}&{}D&{}I\\ \hline \hbox {order-theoretic}&{}D(s)=min\{e\in E:es=s\}&{}I(s)=max\{e\in E:es=e\}\\ \hbox {``Green's relation''}&{}D(s)=e\in E: (e,s)\in \widetilde{{\mathcal R}}_E&{}I(s)=e\in E: (e,s)\in {\mathcal F}_E\\ \hbox {set of axioms}&{}\hbox {finitely based variety}&{}\hbox {finitely based proper quasivariety}\\ \hline \end{array} \end{aligned}$$

3 The $(E,{\mathcal F}_E)$-abundance property

Throughout the remainder, suppose S is a semigroup, with E(S) its set of idempotent elements, and E some subset of E(S).

A simple fact that will be useful from time to time is the following.

Lemma 3.1

For a semigroup S, if $e\in E(S)$ and $s\in S$ are such that $es=e$, then $se\in E(S)$ and $se\sim _r e$.

As noted in the previous section, the most general formulation of the interior operation idea arises via consideration of certain generalised Green-like relations, and we consider these first.

Definition 3.2

Suppose $T\subseteq S$ and $a\in S$. Define ${{\,\textrm{fix}\,}}_T(a)=\{x\in T\mid xa=x\}$; if $T=S$, just write ${{\,\textrm{fix}\,}}(a)$. Dually define ${{\,\textrm{lfix}\,}}_T(a)=\{x\in T\mid ax=x\}$; if $T=S$ we again omit the subscript.

The following is clear.

Proposition 3.3

If non-empty, ${{\,\textrm{fix}\,}}(a)$ is a left ideal of S, and for $T\subseteq S$, ${{\,\textrm{fix}\,}}_T(a)=T\cap {{\,\textrm{fix}\,}}(a)$.

It follows that $(a,b)\in {\mathcal F}$ if and only if ${{\,\textrm{fix}\,}}(a)={{\,\textrm{fix}\,}}(b)$, and that for $E\subseteq E(S)$, $(a,b)\in {\mathcal F}_E$ if and only if ${{\,\textrm{fix}\,}}_E(a)={{\,\textrm{fix}\,}}_E(b)$. The following is now clear.

Proposition 3.4

We have the inclusion ${\mathcal F}\subseteq {\mathcal F}_E$ for any $E\subseteq E(S)$; hence for any $G\subseteq E(S)$, if S is $(G,{\mathcal F})$-abundant, then it is $(G,{\mathcal F}_E)$-abundant.

We show in Example 3.10 below that the converse to the second part of the above is false even if $E=G=E(S)$.

For a semigroup S, denote by $S^0$ the semigroup S with adjoined zero 0. Denote by $T_X$ the transformation semigroup on the non-empty set X. It was shown in [15] that $T_X^0$ is ${\mathcal F}$-abundant, and $T_X$ is ${\mathcal G}$-abundant (whence so is $T_X^0$). Hence by Proposition 3.4, they are ${\mathcal F}_{E(S)}$-abundant and ${\mathcal G}_{E(S)}$-abundant respectively. Note that both $T_X$ and $T_X^0$ are regular; in the regular case, a converse of Proposition 3.4 holds.

Proposition 3.5

Suppose S is regular. Then S is ${\mathcal F}$-abundant if and only if it is ${\mathcal F}_{E(S)}$-abundant.

Proof

Suppose the regular semigroup S is ${\mathcal F}_{E(S)}$-abundant. Then for all $a\in S$, ${{\,\textrm{fix}\,}}_{E(S)}(a)={{\,\textrm{fix}\,}}_{E(S)}(e)$ for some $e\in E(S)$. Hence $fa=f$ if and only if $fe=f$, for all $f\in E(S)$. So the following are equivalent for $b\in S$ (where there is $c\in S$ such that $bcb=b, cbc=c$ by regularity, so that $cb\in E(S)$): $ba=b$; $cba=cb$; $cbe=cb$; $be=b$. Hence ${{\,\textrm{fix}\,}}(a)={{\,\textrm{fix}\,}}(e)$, so S is ${\mathcal F}$-abundant. The converse is immediate from Proposition 3.4. $\square$

Example 3.7 in [15] therefore furnishes an example of a regular (indeed inverse) semigroup which is not ${\mathcal F}_{E(S)}$-abundant, since it was shown there not to be ${\mathcal F}$-abundant.

We have the folowing analog of Proposition 2.1 in [14].

Proposition 3.6

For $e,f\subseteq E(S)$, the following are equivalent:

• 1. $(e,f)\in {\mathcal F}$;

• 2. $(e,f)\in {\mathcal F}_E$ for any E containing e, f;

• 3. $e\sim _r f$.

Proof

If $(e,f)\in {\mathcal F}$ then because ${\mathcal F}\subseteq {\mathcal F}_E$, we have that $(e,f)\in {\mathcal F}_E$ for any E, in particular for any E containing e, f.

If $(e,f)\in {\mathcal F}_E$ for some E containing e, f, then since $ff=f$, we have that $fe=f$ so $f\le _r e$. Symmetry now gives that $e\sim _r f$.

Suppose $e\sim _r f$. If $xe=x$ for some $x\in S$, then because $ef=e$, we obtain that $xf=xef=xe=x$. A similar argument using the fact that $fe=f$ gives that if $xf=x$ for some $x\in S$ then $xe=x$. So $(e,f)\in {\mathcal F}$. $\square$

Corollary 3.7

If $\theta$ is either of ${\mathcal F}$ and ${\mathcal F}_E$, then each $\theta$-class of S has at most one member of E if and only if E is right pre-reduced.

For each $a\in S$, denote by i(a) the (possibly empty) subset of E consisting of the elements of E in the ${\mathcal F}_E$-class of $a\in S$. By Proposition 3.6, these are precisely the elements equivalent under the quasiorder $\le _r$. It follows that $i(S)=\{i(s)\mid s\in S, i(s)\ne \emptyset \}$ is a partially ordered set, with $i(x)\le i(y)$ if and only if $e\le _r f$ for any, hence every, $e\in i(x)$, $f\in i(y)$. This partially ordered set is easily seen to be isomorphic to $E/{\sim _r}$, hence to E itself if E is right pre-reduced.

Proposition 3.8

For all $x\in S$,

$$\begin{aligned}i(x)=\{e\in E\mid ex=e,\hbox { and if }fx=f\hbox { for some }f\in E\hbox { then }f\le _r e\}.\end{aligned}$$

Proof

If $(x,e)\in {\mathcal F}_E$ for $x\in S$ and $e\in E$, then since $ee=e$, it must be that $ex=e$, and if $fx=f$ for $f\in E$, then $fe=f$.

Conversely, suppose $e\in E$ is such that $ex=e$ and $fx=f$ implies $fe=f$ for all $f\in E$. Then for $f\in E$, if $fx=f$ then $fe=f$, while if $fe=f$ then $fx=fex=fe=f$. So $(x,e)\in {\mathcal F}_E$, that is, $e\in i(x)$. $\square$

So i(x) consists of the “largest left zeros" (with respect to $\le _r$) of x in E.

Proposition 3.9

If S is $(E,{\mathcal F})$-abundant, then ${\mathcal F}={\mathcal F}_E$.

Proof

Suppose S is $(E,{\mathcal F})$-abundant. By Proposition 3.4, it suffices to show that ${\mathcal F}_E\subseteq {\mathcal F}$. Suppose $(x,y)\in {\mathcal F}_E$. Since S is $(E,{\mathcal F})$-abundant, there exists $e\in E$ for which $(x,e)\in {\mathcal F}$, so for all $a\in S$, $ax=a$ if and only if $ae=a$. But $ee=e$, so $ex=e$. But since $(x,y)\in {\mathcal F}_E$, this means $ey=e$ as well. If $ax=a$ then $ae=a$, so $ay=aey=ae=a$. Symmetry thus gives equivalence of $ax=a$ with $ay=a$. So $(x,y)\in {\mathcal F}$. $\square$

Example 3.10

An ${\mathcal F}_{E(S)}$-abundant semigroup which is not ${\mathcal F}$-abundant.

Let $T=\{0,a,b,1\}$, made into a semigroup as follows:

$$\begin{aligned} \begin{array}{c|cccc} \cdot &{}0&{}a&{}b&{}1\\ \hline 0&{}0&{}0&{}0&{}0\\ a&{}0&{}0&{}a&{}a\\ b&{}0&{}a&{}1&{}b\\ 1&{}0&{}a&{}b&{}1 \end{array} \end{aligned}$$

(It will follow from Example 5.10 below that this is in fact a semigroup.) Now ${{\,\textrm{fix}\,}}(0)={{\,\textrm{fix}\,}}(a)=\{0\}$, ${{\,\textrm{fix}\,}}(b)=\{0,a\}$, and ${{\,\textrm{fix}\,}}(1)=T$, so ${\mathcal F}$ partitions T as follows: $\{0,a\}, \{b\},\{1\}$, so S is not ${\mathcal F}$-abundant. However, it is ${\mathcal F}_{E(T)}$-abundant, since $E(T)=\{0,1\}$ and ${\mathcal F}_{E(T)}$ partitions T into $\{0,a,b\}$ and $\{1\}$, as is easily checked.

It was shown in [15] that the multiplicative semigroup of a ring R is ${\mathcal F}$-abundant if and only if it is ${\mathcal R}^*$-abundant, if and only if R is a left Baer ring; this extends to right-sided versions of everything (see Theorem 3.3 there). In a very similar way, one can relate $(E,{\mathcal F}_E)$-abundance of the multiplicative semigroup of a ring R to its $(E',\widetilde{{\mathcal R}}_{E'})$-abundance, where $E'=\{1-e\mid e\in E\}$. This then leads to correspondences between generalised D-semigroup structures definable on rings and generalised interior operations definable on them. However, our focus here is on the semigroup case, where the concepts differ.

Analogous to Green’s relation ${\mathcal H}={\mathcal L}\cap {\mathcal R}$ and completely regular semigroups, in [15] we defined ${\mathcal K}={\mathcal F}\cap {\mathcal G}$.

Definition 3.11

For a semigroup S, with $E\subseteq E(S)$, define ${\mathcal K}_E={\mathcal F}_E\cap {\mathcal G}_E$.

Obviously, if S is ${\mathcal K}$-abundant then it is both ${\mathcal F}$-abundant and ${\mathcal G}$-abundant, while if it is $(E,{\mathcal K}_E)$-abundant then it is both $(E,{\mathcal F}_E)$-abundant and $(E,{\mathcal G}_E)$-abundant. Of course, ${\mathcal K}$-abundance is analogous to the notions of complete regularity, superabundance and U-superabundance, respectively defined using Green’s relations ${\mathcal L}$ and ${\mathcal R}$ (with ${\mathcal H}={\mathcal L}\cap {\mathcal R}$), the generalised Green’s relations ${\mathcal L}^*$ and ${\mathcal R}^*$ (with ${\mathcal H}^*={\mathcal L}^*\cap {\mathcal R}^*$ as in [3]), and the still further generalised Green’s relations $\widetilde{\mathcal L}_E$ and $\widetilde{\mathcal R}_E$ where $E\subseteq E(S)$ (with $\widetilde{\mathcal H}_E=\widetilde{\mathcal L}_E\cap \widetilde{\mathcal R}_E$ as in [12] and then [9]). In contrast to completely regular semigroups, in which each ${\mathcal H}$-class is a group, or superabundant semigroups, in which each ${\mathcal H}^*$-class is a cancellative semigroup of a special kind, the ${\mathcal K}$-classes of a ${\mathcal K}$-abundant semigroup are not even subsemigroups in general, as shown in Example 4.2 of [15].

From Proposition 3.6 and its dual, and the fact that for idempotents e, f, if $e\sim _r f$ and $e\sim _l f$, then $e=f$, we have the following (the first part of which was noted in [15]).

Proposition 3.12

If S is a semigroup and $E\subseteq E(S)$, then each ${\mathcal K}$-class contains at most one idempotent and each ${\mathcal K}_E$-class contains at most one idempotent from E. Hence if S is $(E,{\mathcal K})$-abundant then $E=E(S)$ and so it is ${\mathcal K}$-amiable, and if it is $(E,{\mathcal K}_E)$-abundant then it is $(E,{\mathcal K}_E)$-amiable.

It was noted in Proposition 5.3 of [15] that if a semigroup is ${\mathcal F}$-abundant, then it is both ${\mathcal F}$-amiable and ${\mathcal G}$-amiable if and only if ${\mathcal F}={\mathcal G}$. This has the following variant.

Proposition 3.13

Suppose S is an ${\mathcal F}_{E(S)}$-abundant semigroup. Then S is ${\mathcal F}_{E(S)}$-amiable and ${\mathcal G}_{E(S)}$-amiable if and only if ${\mathcal F}_{E(S)}={\mathcal G}_{E(S)}$.

Proof

The proof follows that of Proposition 5.3 in [15] (plus that of Lemma 5.2 and its dual in [15]) with few differences, but for completeness we present it afresh here. So assume S is ${\mathcal F}_{E(S)}$-abundant.

If ${\mathcal F}_{E(S)}={\mathcal G}_{E(S)}$ in S, then trivially S is ${\mathcal G}_{E(S)}$-abundant as well. But if $e,f\in E(S)$ and $(e,f)\in {\mathcal F}_{E(S)}={\mathcal G}_{E(S)}$ then $e\sim _r f$ and $e\sim _l f$ by Proposition 3.6 and its dual, so $e=f$. So S is ${\mathcal F}_{E(S)}$-amiable and ${\mathcal G}_{E(S)}$-amiable.

Conversely, suppose S is ${\mathcal F}_{E(S)}$-amiable and ${\mathcal G}_{E(S)}$-amiable. Pick e in E(S) and $s\in S$ such that $(s,e)\in {\mathcal F}_{E(S)}$, and let $f\in E(S)$ be such that $(s,f)\in {\mathcal G}_{E(S)}$. Then $es=e$ (since $ee=e$ and $(s,e)\in {\mathcal F}_{E(S)}$), so $se\in E(S)$ and $e\sim _r se$ by Lemma 3.1, and so by Proposition 3.6, $se=e$; dually, $fs=f$. But $(s,f)\in {\mathcal G}_{E(S)}$, so $se=e$ implies that $fe=e$, but also $(s,e)\in {\mathcal F}_{E(S)}$ so $fs=f$ implies that $fe=f$. Hence $e=fe=f$. It follows that $(s,e)\in {\mathcal F}_{E(S)}$ implies $(s,e)\in {\mathcal G}_{E(S)}$. Since this is true for all idempotents e, ${\mathcal F}_{E(S)}\subseteq {\mathcal G}_{E(S)}$. By symmetry, they are equal. $\square$

4 Left GI-semigroups

Throughout, let S be a semigroup. If T is any unary operation on S, we denote by T(S) the subset $\{T(s)\mid s\in S\}$ of S.

As indicated in the introduction, we make the following formal definition.

Definition 4.1

Let S be a unary semigroup with unary operation I such that $I(s)\in E(S)$ for all $s\in S$. We say S is a left GI-semigroup (a left generalised interior semigroup) if S is $(I(S),{\mathcal F}_{I(S)})$-amiable. We say S is an $F^I$ -abundant left GI-semigroup if it is $(I(S),{\mathcal F})$-amiable (in which case ${\mathcal F}_{I(S)}={\mathcal F}$ by Proposition 3.9 and so S is indeed a left GI-semigroup). We say the left GI-semigroup S is central if I(S) consists of central idempotents.

In [15], $F^I$-abundant left GI-semigroups were considered in the case where $I(S)=E(S)$ and first-order axioms were given; see Proposition 5.8 there. (We note an error in the laws in that result: the law $F(s)s=s$ should be $F(s)s=F(s)$. The proof given there requires an equally small adjustment. There are similar corrections needed to the statement of Proposition 4.3 of [15] and its proof.) We generalise that here.

Clearly, if a left GI-semigroup is $F^I$-abundant then it is ${\mathcal F}$-abundant as a semigroup, and a central left GI-semigroup is also a right GI-semigroup under the same unary operation.

Proposition 4.2

Let S be a semigroup with $E\subseteq E(S)$.

If S is $(E,{\mathcal F}_E)$-abundant (resp. $(E,{\mathcal F})$-abundant) then there exists a left GI-semigroup structure (resp. an $F^I$-abundant left GI-semigroup structure) on S which is such that $I(S)\subseteq E$, and $(a,b)\in {\mathcal F}_E$ (resp. $(a,b)\in {\mathcal F}$) if and only if $I(a)=I(b)$. Namely, pick one $e\in E$ from each $\sim _r$-class to give $F\subseteq E$, and define $I(x)=e$ where $e\in F$ is such that $(e,x)\in {\mathcal F}_E$ (resp. $(e,x)\in {\mathcal F}$).

Proof

Suppose S is $(E,{\mathcal F}_E)$-abundant. Clearly S is $(I(S),{\mathcal F}_{I(S)})$-amiable if F and I are defined as stated, and is a left GI-semigroup when I is added to the signature. It remains to check that ${\mathcal F}_{I(S)}\subseteq {\mathcal F}_E$, the opposite inclusion being obvious.

Suppose $(x,y)\in {\mathcal F}_{I(S)}$. Then for all $g\in I(S)$, $gx=g$ if and only if $gy=g$. Suppose $ex=e$ for some $e\in E$. Then $I(e)e=I(e)$. But also, I is defined in such a way that $(e,I(e))\in {\mathcal F}_E$, so by Proposition 3.6, $eI(e)=e$ and $I(e)e=I(e)$. Hence, $I(e)x=I(e)ex=I(e)e=I(e)$, so $I(e)y=I(e)$, and so $ey=eI(e)y=eI(e)=e$. By symmetry, $ex=e$ if and only if $ey=e$, for all $e\in E$, and so $(x,y)\in {\mathcal F}_E$, as required.

The second part follows from Proposition 3.9. $\square$

The following is clear.

Proposition 4.3

For the semigroup S and $E\subseteq E(S)$, the following are equivalent.

• 1. S is a left GI-semigroup with $I(S)=E$.

• 2. S is $(E,{\mathcal F}_E)$-abundant and E is right pre-reduced.

Similarly, the following are equivalent.

• 1. S is an $F^I$-abundant left GI-semigroup with $I(S)=E$.

• 2. S is $(E,{\mathcal F})$-abundant and E is right pre-reduced.

• 3. S is $(E,{\mathcal F})$-abundant and E is maximal right pre-reduced in E(S).

One natural choice of right pre-reduced set of idempotents in a semigroup includes its set of central idempotents $C(S)=\{e\in E(S)\mid es=se\hbox { for all }s\in S\}$. In the case of an involuted semigroup, another natural choice is its set of projections (symmetric idempotents) $E^*(S)=\{e\in E(S)\mid e^*=e\}$; this is I(R) in a Rickart $*$-ring, as described in the introduction, and such cases are $(E^*(R),{\mathcal F})$-abundant, so $E^*(R)$ is maximal right pre-reduced in E(R) by the above result. In both these cases $\le _r$ coincides with the natural order on E.

As previewed in Sect. 2, there is an order-theoretic characterisation of left GI-semigroups.

Proposition 4.4

Let S be a semigroup with $E\subseteq E(S)$. If for all $x\in S$ there is a unique largest $e\in E$ under $\le _r$ for which $es=e$, then S is a left GI-semigroup in which I(s) is this largest element for all s. Conversely, if S is a left GI-semigroup then $I(S)\subseteq E(S)$ is such that for all $s\in S$, I(s) is the unique largest $e\in I(S)$ under $\le _r$ for which $es=e$. In this case $\le _r$ is a partial order on I(S); equivalently, I(S) is right pre-reduced.

Proof

Suppose S is a semigroup with $E\subseteq E(S)$ such that for all $x\in S$ there is a unique largest $e\in E$ under $\le _r$ for which $ex=e$. Then for such x, by Proposition 3.8, i(x) contains a single element. So every ${\mathcal F}_E$-class contains a unique element of E, and so S is a left GI-semigroup if we define $I(s)=e$ (where $(s,e)\in {\mathcal F}_E$) and do the same for each $s\in S$, and then $I(S)=E$.

Conversely, if S is a left GI-semigroup then for all $x\in S$, i(x) contains precisely one element $e\in I(S)$, namely I(x), which has the given description by Proposition 3.8.

If $e,f\in E$ and $ef=e$ and $fe=f$ then $e\sim _r f$, so by Proposition 3.6, $(e,f)\in {\mathcal F}_E$, and so $e=f$. Hence E is right pre-reduced and so $\le _r$ is a partial order on E. $\square$

Of particular interest are cases in which the natural order on E may be used to define I, which is the original motivation for the study of such operations, discussed in Sect. 2.

Definition 4.5

Let S be a unary semigroup with unary operation I. We say it is a left interior semigroup if $I(S)=\{I(s)\mid s\in S\}$ consists of idempotents, and for all $s\in S$, I(s) is the largest (under the natural order) $e\in I(S)$ for which $es=e$.

We have the following consequence of Proposition 4.4.

Corollary 4.6

Let S be a unary semigroup with unary operation I. Then S is a left interior semigroup if and only if it is a left GI-semigroup in which I(S) is right reduced.

Proof

Suppose S is a left interior semigroup. Then for $e,f\in I(S)$, suppose $e=ef$. Then $e\le I(f)$. But $ff=f$ so $f\le I(f)$, yet $I(f)f=I(f)$, and so $f=I(f)f=I(f)$. Hence $e\le f$, so $e=fe$. So I(S) is right reduced, and so $\le _r$ equals $\le$ on I(S). It now follows from Proposition 4.4 that S is a left GI-semigroup.

Conversely, suppose S is a left GI-semigroup and I(S) is right reduced, so that $\le _r$ equals $\le$ on I(S). By Proposition 4.4, I(s) is the unique largest $e\in I(S)$ under $\le _r$ for which $es=e$, hence is the largest under $\le$ with this property. Hence, S is a left interior semigroup. $\square$

From the above and Proposition 4.3, we now obtain the following.

Corollary 4.7

For the semigroup S and $E\subseteq E(S)$, the following are equivalent.

• 1. S is a left interior semigroup with $I(S)=E$.

• 2. S is $(E,{\mathcal F}_E)$-abundant and E is right reduced.

A third way to view left GI-semigroups is via first order logic, in terms of equations and quasiequations.

Theorem 4.8

Let S be a semigroup. If S is a left GI-semigroup, then the following laws are satisfied by I:

• $I(x)x=I(x)$;

• $I(x)y=I(x) \Rightarrow I(x)I(y)=I(x)$;

• $(\; I(x)I(y)=I(x)\ \& \ I(y)I(x)=I(y)\; ) \Rightarrow I(x)=I(y)$.

Conversely, if S is a unary semigroup satisfying the above laws, then it is a left GI-semigroup with respect to I(S).

Amongst left GI-semigroups, the class of left interior semigroups is specified by the additional law

• $I(x)I(y)=I(x)\Rightarrow I(y)I(x)=I(x)$.

The left GI-semigroup S is $F^I$-abundant if and only if it satisfies the law

• $xy=x \Rightarrow xI(y)=x$.

All of the above classes of unary semigroups are proper quasivarieties.

Proof

If S is a left GI-semigroup then the above laws all hold by Proposition 4.3.

Conversely, assume the above laws hold. First note that because $I(x)x=I(x)$, then by the second law, $I(x)I(x)=I(x)$. So $I(x)\in E(S)$ for all $x\in S$. The first two laws guarantee that I(x) is a largest element of I(S) under the quasiorder $\le _r$ for which $ex=e$, and the third ensures it is the unique largest, so S is a left GI-semigroup by Proposition 4.4.

The claim regarding the class of $F^I$-abundant left GI-semigroups follows from the fact that the stated law expresses the fact that $(y,I(y))\in {\mathcal F}$ for all $y\in S$, establishing $(I(S),{\mathcal F})$-abundance, the rest following from the fact that I(S) is right pre-reduced.

To show both classes are proper quasivarieties, consider the following semigroup: $S=\{a,e,f,g,1\}\subseteq T(X)$, the full transformation semigroup on X, where $X=\{x,y,z\}$ and

$$\begin{aligned}{} & {} a=\{(x,y),(y,x),(z,z)\}, e=\{(x,x),(y,x),(z,x)\}, f=\{(x,y),(y,y),(z,y)\},\\ {}{} & {} g=\{(x,z),(y,z),(z,z)\}, 1=\{(x,x),(y,y),(z,z)\}.\end{aligned}$$

It is easily checked that this is a subsemigroup of T(X) with the following multiplication table:

$$\begin{aligned} \begin{array}{c|ccccc} \cdot &{}a&{}e&{}g&{}f&{}1\\ \hline a&{}1&{}e&{}g&{}f&{}a\\ e&{}g&{}e&{}g&{}f&{}e\\ g&{}e&{}e&{}g&{}f&{}g\\ f&{}f&{}e&{}g&{}f&{}f\\ 1&{}a&{}e&{}g&{}f&{}1 \end{array} \end{aligned}$$

Let $E=E(S)=S\backslash \{a\}$. Then ${{\,\textrm{fix}\,}}_E(a)=\{f\}={{\,\textrm{fix}\,}}_E(f)$, while ${{\,\textrm{fix}\,}}_E(e)=\{e\}, {{\,\textrm{fix}\,}}_E(g)=\{g\}$ and ${{\,\textrm{fix}\,}}_E(1)=E$, so ${\mathcal F}_E$ partitions S as follows: $\{a,f\},\{e\},\{g\},\{1\}$, so S is $(E,{\mathcal F}_E)$-abundant and so can be made into a left GI-semigroup with $I(S)=E$.

There is a congruence on S that respects I and which partitions S into $\{e,g\}$ and the three singleton sets $\{a\},\{f\},\{1\}$. Representing the singleton sets using their only elements and writing e for $\{e,g\}$ gives the following quotient.

$$\begin{aligned} \begin{array}{c|cccc} \cdot &{}a&{}e&{}f&{}1\\ \hline a&{}1&{}e&{}f&{}a\\ e&{}e&{}e&{}f&{}e\\ f&{}f&{}e&{}f&{}f\\ 1&{}a&{}e&{}f&{}1 \end{array} \end{aligned}$$

Now $I(a)=f$ with $I(x)=x$ for all other x, and so we observe that $ea=e$ yet $eI(a)=ef=f$; hence the quotient is not a left GI-semigroup since the fourth law in Proposition 4.10 fails.

Note that E in the above example is right reduced, so S is a left interior semigroup. $\square$

The laws for ${\mathcal F}$-amiable semigroups (viewed as left GI-semigroups) given in Proposition 5.8 of [15] (again noting the error there) are the above laws for $F^I$-abundant semigroups, with the second law for left GI-semigroups removed since it follows from the additional law for the $F^I$-abundant case, together with a further law that forces I(S) to be all of E(S), namely “$x^2=x\Rightarrow I(x)=x$".

Lemma 4.9

If S is a left GI-semigroup then for all $x,y\in S$, the laws $I(I(x))=I(x)$ and $I(xI(yI(x)))=I(yI(x))$ hold.

Proof

Now for each $x\in S$, I(x) is the unique member of I(S) in the ${\mathcal F}_E$-class containing I(x), which is of course I(x) itself, so $I(x)=I(I(x))$.

By the first law,

$$\begin{aligned}{} & {} I(xI(yI(x)))yI(x)=I(xI(yI(x)))xI(yI(x))yI(x)\\{} & {} =I(xI(yI(x)))xI(yI(x))=I(xI(yI(x))), \end{aligned}$$

and so by the second law, $I(xI(yI(x)))I(yI(x))=I(xI(yI(x)))$. On the other hand, again using the first law,

$$\begin{aligned}\begin{aligned}&I(yI(x))xI(yI(x))=I(yI(x))yI(x)xI(yI(x))\\ {}&=I(yI(x))yI(x)I(yI(x))=I(yI(x))I(yI(x))=I(yI(x)),\end{aligned}\end{aligned}$$

so by the second, $I(yI(x))I(xI(yI(x)))=I(yI(x))$. Now apply the third law to give equality of I(yI(x)) and I(xI(yI(x))). $\square$

In fact the two equational laws obtained in Lemma 4.9 may be used to replace the third law for left GI-semigroups, leaving only one quasiequation.

Proposition 4.10

The laws for left GI-semigroups may be equivalently given as semigroups with unary operation I satisfying the following laws:

• $I(x)x=I(x)$

• $I(I(x))=I(x)$

• $I(xI(yI(x)))=I(yI(x))$

• $I(x)y=I(x) \Rightarrow I(x)I(y)=I(x)$.

Proof

By Lemma 4.9 the class of algebras satisfying the above four laws contains the class of left GI-semigroups. Conversely, consider a unary semigroup satisfying the given laws. Suppose $I(x)I(y)=I(x)$ and $I(y)I(x)=I(y)$. Then using the third law above, with x replaced by I(x) and y replaced by I(y) gives $I(I(x)I(I(y)I(I(x))))=I(I(y)I(I(x)))$, so using the second law above we obtain $I(I(x)I(I(y)I(x)))=I(I(y)I(x))$, and then using our assumption about x, y, we obtain $I(I(x)I(I(y)))=I(I(y))$, so $I(I(x)I(y))=I(y)$, and so $I(I(x))=I(y)$, so $I(x)=I(y)$, so the third left GI-semigroup law holds. $\square$

There is an easy corollary to Proposition 4.10 and the final part of Proposition 4.8, applying to the $F^I$-abundant case.

Corollary 4.11

The laws for $F^I$-abundant left GI-semigroups may be equivalently given as semigroups with unary operation I satisfying the following laws:

• $I(x)x=I(x)$

• $I(I(x))=I(x)$

• $I(xI(yI(x)))=I(yI(x))$

• $xy=x \Rightarrow xI(y)=x$.

Not all left GI-semigroups are $F^I$-abundant.

Proposition 4.12

Suppose S is a semilattice which is a left GI-semigroup. Then it is $F^I$-abundant if and only if $I(x)=s$ for all $s\in S$.

Proof

In a semilattice, $\sim _r$ is the equality relation, so every element is in a distinct ${\mathcal F}$-class by Proposition 3.6. $\square$

So any semilattice that is a left GI-semigroup in a non-trivial way (with $I(s)\ne s$ for some s) cannot be $F^I$-abundant. Examples arise from the subsets of a topological space, where I is the usual interior operator on subsets. Note also that every semilattice (indeed every band) is trivially ${\mathcal F}$-abundant, showing that for a left GI-semigroup S, ${\mathcal F}$-abundance of S as a semigroup is not sufficient to ensure its $F^I$-abundance as a left GI-semigroup.

As noted earlier, a non-trivial example of a left GI-semigroup comes from the theory of partial function semigroups, where I(s) is the identity map on the subset $\{x\in {{\,\textrm{dom}\,}}(s)\mid xs=x\}$. Then the law $st=s \Rightarrow sI(t)=s$ is easily seen to hold, so this example is $F^I$-abundant. It was shown in [6] that the class of all such functional unary semigroups has no finite axiomatisation.

The case of associative rings whose multiplicative semigroups are interior semigroups in the earlier sense in which I(S) is a subsemigroup which is a semilattice was considered in [8]. A different approach is as follows. On any associative ring, one may define the so-called adjoint (or circle) operation $\circ$ by setting $a\circ b=a+b-ab$ for all $a,b\in R$. Then $(R,\circ )$ is a semigroup. (This operation is due to Nathan Jacobson and was essential in his study of radicals of associative rings; see [7].) Note that the adjoint operation exchanges the closure and interior operations, so that in particular, (R, .) is a generalised D-semigroup if and only if $(R,\circ )$ is a left GI-semigroup (and dually). Hence, the closure rings considered in [4] are nothing but associative rings equipped with a unary operation making the semigroup $(R,\circ )$ (the opposite of) an interior semigroup in the above sense.

Bands are important types of semigroups, and for these we obtain an equational decription of left GI-semigroups.

Proposition 4.13

The class of all left GI-semigroups in which S is a band is a variety, in which the equation $x^2=x$ is added and the quasiequation in Proposition 4.10 may be replaced by the equation $I(xy)I(y)=I(xy)$.

Proof

Suppose the left GI-semigroup S is a band. Then for all $x,y\in S$, we have that $I(xy)y=I(xy)xyy=I(xy)xy=I(xy)$, so by the quasiequation in Proposition 4.10, $I(xy)I(y)=I(xy)$.

Conversely, suppose S is a unary band satisfying the equational laws in Proposition 4.10 together with $I(xy)I(y)=I(xy)$. If $x,y\in S$ are such that $I(x)y=I(x)$, then because $I(I(x)y)I(y)=I(I(x)y)$, we have that $I(I(x))I(y)=I(I(x))$, so $I(x)I(y)=I(x)$, and so S is a left GI-semigroup. $\square$

By contrast, the class of left GI-semigroups in which I(S) is a band (an assumption corresponding to the equation $I(I(x)I(y))=I(x)I(y)$) is not a variety, since the example in the proof of Theorem 4.8 has this property but its homomorphic image is not a left GI-semigroup.

Proposition 4.14

The class of all $F^I$-abundant left GI-semigroups in which S is a band is a variety, in which the quasiequation in Corollary 4.11 and the band law $x^2=x$ may be replaced by the equation $xI(x)=x$.

Proof

If the $F^I$-abundant left GI-semigroup S is a band, then for all $x\in S$, because $x^2=x$, we have $xI(x)=x$. Conversely, if S is a unary semigroup satisfying the equational laws in Proposition 4.10 together with $xI(x)=x$, then if $x,y\in S$ are such that $xy=x$, we have $xI(y)=xyI(y)=xy=x$, and $x^2=xI(x)x=xI(x)=x$. $\square$

Corollary 4.15

If S is a central $F^I$-abundant left GI-semigroup which is a band, then S is a semilattice with $I(s)=s$ for all $s\in S$.

Proof

By Proposition 4.14, for all $x\in S$ we have $I(x)=I(x)x=xI(x)=x$, and so $S=I(S)$ is a semilattice. $\square$

Example 5.10 shows that in general, the class of central left GI-semigroups is not a variety, even in the $F^I$-abundant case. However, in other respects they are well-behaved at least in the $F^I$-abundant case.

Proposition 4.16

If S is a central $F^I$-abundant left GI-semigroup, then we have that $I(xI(y))=I(x)I(y)$ for all $x,y\in S$, so I(S) is a subsemigroup of S.

Proof

First note that $I(x)I(y)=I(x)I(y)xI(y)$, so

$$\begin{aligned} I(x)I(y)= & {} I(x)I(y)I(xI(y))\\= & {} I(xI(y))I(x)I(y)\\= & {} I(xI(y))xI(y)I(x)I(y)\\= & {} I(xI(y))xI(y)I(x)\\= & {} I(xI(y))I(x). \end{aligned}$$

But also,

$$\begin{aligned} I(xI(y))x= & {} I(xI(y))xI(y)x\\= & {} I(xI(y))xI(y)xI(y)\\= & {} I(xI(y))xI(y)\\= & {} I(xI(y)), \end{aligned}$$

so $I(xI(y))=I(xI(y))I(x)=I(x)I(y)$. $\square$

5 Meets in I(S)

The interior semigroups of [8] (more precisely, their left-sided duals) are defined to be unary semigroups given by the following equational laws:

• $I(x)I(y)=I(y)I(x)$

• $I(x)x=I(x)$

• $I(I(x))=I(x)$

• $I(x)I(xy)=I(x)I(y)$.

In [8], it was shown that these are nothing but semigroups equipped with a subsemilattice E such that for all $x\in S$, I(x) is the largest $e\in E$ for which $ex=e$, and then $E=I(S)$. So these are certainly left GI-semigroups as well, and it is obvious that they may alternatively be described as left GI-semigroups such that for all $x,y\in S$, $I(x)I(y)=I(y)I(x)=I(I(x)I(y))$, since these extra laws ensure that I(S) not only commutes but is a semilattice.

In such an interior semigroup, ef is of course the meet, or greatest lower bound, of $e,f\in I(S)$, with respect to the natural order (which in this case equals $\le _r$). However, in general left GI-semigroups, such meets need not exist at all: any right zero semigroup in which $I(x)=x$ for all x gives a left GI-semigroup, as is easily checked; however, meets do not exist since $\le _r$ is just equality.

It was shown in Proposition 5.9 of [15] that in an $F^I$-abundant left GI-semigroup S in which $I(S)=E(S)$ is reduced (rather than just right reduced, as it must be), E(S) is a meet-semilattice under the natural order, with $e\wedge f=I(ef)$ for all $e,f\in E(S)$. In general, we have the following.

Proposition 5.1

Let S be a left GI-semigroup. Then for any $e,f\in I(S)$, I(ef) is an upper bound on the set of common lower bounds of e and f, and $I(ef)\le _r f$. Hence, I(ef) is the greatest lower bound of $e,f\in I(S)$ if and only if $I(ef)\le _r e$.

Proof

First, suppose $e,f,g\in I(S)$ with $g\le _r e$ and $g\le _r f$. So $ge=g=gf$. Then $g(ef)=gf=g$, so $gI(ef)=g$, and so $g\le _r I(ef)$. Secondly, $I(ef)f=I(ef)eff=I(ef)ef=I(ef)$, so $I(ef)\le _r f$. The final claim is now immediate. $\square$

So in any left GI-semigroup S, if $e,f\in I(S)$ have a greatest lower bound with respect to $\le _r$, it is at most I(ef).

Definition 5.2

Let S be a left GI-semigroup in which, for all $e,f\in I(S)$, I(ef) is their meet with respect to $\le _r$. Then we say S has large meets.

Thus, $F^I$-abundant left GI-semigroups in which $I(S)=E(S)$ is reduced have large meets, by Proposition 5.9 in [15]. This carries over to left GI-semigroups in general, as we now show. (The proof is quite similar to that of Proposition 5.9 in [15] but we include it here for completeness.)

Proposition 5.3

Suppose S is a left GI-semigroup in which $I(S)=E(S)$ is reduced. Then S has large meets.

Proof

For any $e,f\in I(S)$, we have that $I(ef)ef=ef$, so by Lemma 3.1, $efI(ef)\in E(S)$ and indeed $efI(ef)\sim _r I(ef)$, so by the reduced property, $efI(ef)=I(ef)$, so $eI(ef)=I(ef)$ and so $I(ef)\le _l e$, so $I(ef)\le _r e$ by the reduced property. Hence by Proposition 5.1, I(ef) is the greatest lower bound of e, f. So S has large meets. $\square$

The assumption that $I(S)=E(S)$ in the above cannot be dropped, as we shall see later.

In general, those left GI-semigroups having large meets have a useful and relatively simple characterisation.

Proposition 5.4

Let S be a left GI-semigroup. Then S has large meets if and only if it satisfies the law $I(ef)=I(fe)$ for all $e,f\in I(S)$. In this case, I(S) is left reduced and satisfies the law $I(I(ey)e)I(y)=I(I(ey)e)$ for all $e\in I(S)$ and $y\in S$.

Proof

If I(ef) is the meet of every $e,f\in I(S)$ under $\le _r$, this forces $I(ef)=I(fe)$ for all such e.f. On the other hand, if $I(ef)=I(fe)$ for some $e,f\in I(S)$, then

$$\begin{aligned}I(ef)e=I(fe)e=I(fe)fee=I(fe)fe=I(fe)=I(ef),\end{aligned}$$

and so $I(ef)\le _r e$, and so by Proposition 5.1, I(ef) is the meet of e, f with respect to $\le _r$. So S has large meets if and only if $I(ef)=I(fe)$ for all $e,f\in I(S)$.

So if S has large meets and $e,f\in I(S)$ are such that $ef=f$, then $I(fe)=I(ef)=I(f)=f$, so $fe=ffe=I(fe)fe=I(fe)=f$. Hence, I(S) is left reduced. Finally, for all $e\in I(S)$ and $y\in S$, $I(I(ey)e)y=I(I(ey)e)I(ey)ey=I(I(ey)e)I(ey)=I(I(ey)e)$ since $I(I(ey)e)\le _r I(ey)$ by the large meets property, and so $I(I(ey)e)I(y)=I(I(ey)e)$. $\square$

Corollary 5.5

Every left GI-semigroup in which I(S) commutes has large meets.

The converse of the above fails in general, as we see later. Next we note that the class of left GI-semigroups with large meets can be defined equationally.

Proposition 5.6

The class of left GI-semigroups with large meets is the variety of unary semigroups given by the following laws.

• 1. $I(x)x=I(x)$

• 2. $I(I(x))=I(x)$

• 3. $I(I(I(x)y)I(x))I(y)=I(I(I(x)y)I(x))$.

• 4. $I(I(x)I(y))=I(I(y)I(x))$

Proof

The stated laws hold in any left GI-semigroup with large meets by Lemma 4.9 and Proposition 5.4. Conversely, suppose a left GI-semigroup S satisfies the above four laws. Now if $I(a)b=I(a)$ for $a,b\in S$, then letting $x=a$ and $y=b$ in the third law gives that $I(I(I(a)b)I(a))I(b)=I(I(I(a)b)I(a))$; but $I(I(I(a)b)I(a))=I(I(I(a))I(a))=I(I(a)I(a))=I(I(a))=I(a)$, and so we obtain $I(a)=I(a)I(b)$. Moreover if $ef=e$ and $fe=f$ (where $e,f\in I(S)$), then $e=I(e)=I(ef)=I(fe)=I(f)=f$. So by Proposition 4.8, S is a left GI-semigroup. The final law now gives that S has large meets upon using Proposition 5.4. $\square$

Corollary 5.7

The class of left GI-semigroups in which I(S) commutes is a variety of unary algebras.

However, classes of $F^I$-abundant left GI-semigroups in which even quite strong conditions are placed on I(S) are not so well-behaved.

Example 5.8

An $F^I$-abundant left GI-semigroup in which I(S) is a semilattice under multiplication with a quotient that is not $F^I$-abundant.

Recall Example 3.8 in [15]. It is the commutative monoid with zero $S=\{a,b,c,0,1\}$ with multiplication table as follows:

$$\begin{aligned} \begin{array}{c|ccccc} \cdot &{}a&{}b&{}c&{}0&{}1\\ \hline a&{}0&{}0&{}b&{}0&{}a\\ b&{}0&{}0&{}a&{}0&{}b\\ c&{}b&{}a&{}1&{}0&{}c\\ 0&{}0&{}0&{}0&{}0&{}0\\ 1&{}a&{}b&{}c&{}0&{}1 \end{array} \end{aligned}$$

Now $E(S)=\{0,1\}$, and it was noted that the ${\mathcal F}$-classes of S are $\{0,a,b,c\}$ and $\{1\}$, so S is ${\mathcal F}$-amiable, so can be viewed as an $F^I$-abundant left GI-semigroup in which $I(S)=E(S)$. In Example 3.9 in [15], the semigroup congruence $\theta$ on S was defined by setting $(a,b)\in \theta$, with all other pairs of distinct elements unrelated, and is easily seen to respect I as well. But the quotient $S'=S/\theta$ was observed in [15] to not even be ${\mathcal F}$-abundant, so certainly the quotient left GI-semigroup structure on $S'$ cannot be $F^I$-abundant.

So, in contrast to the class of all left GI-semigroups, the subclass of $F^I$-abundant left GI-semigroups is a proper quasivariety. This example also shows that I(S) being a subsemilattice of S is not enough to ensure homomorphic closure of $F^I$-abundant left GI-semigroups.

Example 5.9

A left GI-semigroup which is a semilattice but in which I(S) is not even a subsemigroup of S.

Consider the poset $S=\{e,f,a,0\}$, in which $0\le a\le e,f$, a meet-semilattice with multiplication table as follows.

$$\begin{aligned} \begin{array}{c|cccc} \cdot &{}e&{}f&{}a&{}0\\ \hline e&{}e&{}a&{}a&{}0\\ f&{}a&{}f&{}a&{}0\\ a&{}a&{}a&{}a&{}0\\ 0&{}0&{}0&{}0&{}0 \end{array} \end{aligned}$$

Let $E=\{e,f,0\}$. Then ${{\,\textrm{fix}\,}}_E(e)=\{e\}$, ${{\,\textrm{fix}\,}}_E(f)=\{f\}$, and ${{\,\textrm{fix}\,}}_E(a)={{\,\textrm{fix}\,}}_E(0)=\{0\}$, so S is a left GI-semigroup with $I(a)=0$ and with all other elements fixed by I. Note that $ef=fe=a$, so I(S) is not a band. Yet by Corollary 5.5, I(S) is a semilattice under $\le$ in which $g\wedge h=I(gh)$ for all $g,h\in I(S)$, and indeed $e\wedge f=I(ef)=I(a)=0$. Contrast this with Corollary 4.15 for the $F^I$-abundant case.

On the other hand, not all left GI-semigroups in which meets exist in I(S) have large meets, even in the $F^I$-abundant case. The multiplicative subsemigroup of the Rickart $*$-ring R is an $F^I$-abundant left GI-semigroup in which $I(R)=E^*(R)$ is a lattice with respect to $\le _r$ (which is the natural order on $E^*(S)$ since it is reduced), but not one in which $e\wedge f=I(ef)$ in general.

Example 5.10

A central $F^I$-abundant left GI-semigroup having a non-$F^I$-abundant left GI-semigroup quotient.

Let $X=\{x,y,z\}$ and let $S=\{a,b,c,0,1\}\subseteq PT(X)$ where $a=\{(x,y)\}$, $b=\{(x,x),(y,z),(z,y)\}$, $c=\{(x,z)\}$, 0 is the empty function and 1 is the identity function on X. Then S is a semigroup, having the following multiplication table.

$$\begin{aligned} \begin{array}{c|ccccc} \cdot &{}0&{}a&{}b&{}c&{}1\\ \hline 0&{}0&{}0&{}0&{}0&{}0\\ a&{}0&{}0&{}c&{}0&{}a\\ b&{}0&{}a&{}1&{}c&{}b\\ c&{}0&{}0&{}a&{}0&{}c\\ 1&{}0&{}a&{}b&{}c&{}1 \end{array} \end{aligned}$$

Then $E(S)=\{0,1\}$, and because $xy=x$ if and only if $x=0$ or $y=1$, ${\mathcal F}$ partitions S into $\{0,a,b,c\}$ and $\{1\}$, so S is $(\{0,1\},{\mathcal F})$-abundant, and so may be viewed as an $F^I$-abundant left GI-semigroup in which $I(1)=1$, with $I(x)=0$ otherwise. In this example, I(S) has the strongest possible properties: its elements are central and form a subsemilattice of S, so $e\wedge f=ef$ for all $e,f\in I(S)$. However, there is a homomorphic image of S which, although a left GI-semigroup by Corollary 5.7, is not $F^I$-abundant. Define $\theta$ by setting $(a,c)\in \theta$, with all other elements unrelated. This is easily checked to be a congruence respecting I, and (writing the congruence class $\{a,c\}$ as a, and so on) $S'=S/\theta$ has multiplication table as follows:

$$\begin{aligned} \begin{array}{c|cccc} \cdot &{}0&{}a&{}b&{}1\\ \hline 0&{}0&{}0&{}0&{}0\\ a&{}0&{}0&{}a&{}a\\ b&{}0&{}a&{}1&{}b\\ 1&{}0&{}a&{}b&{}1 \end{array} \end{aligned}$$

This is Example 3.10, which is not ${\mathcal F}$-abundant, hence certainly cannot be $F^I$-abundant as a left GI-semigroup.

6 Double GI-semigroups

A given semigroup may be both a left GI-semigroup and a right GI-semigroup in independent ways. However, sometimes the same set of idempotents is involved in both.

Definition 6.1

We say S is a double GI-semigroup (resp. double interior semigroup) if it is a left GI-semigroup (resp. left interior semigroup) with unary operation I and a right GI-semigroup (resp. right interior semigroup) with unary operation J, and $I(S)=J(S)$. We say it is $FG^I$-abundant if it is $F^I$-abundant as a left GI-semigroup and $G^I$-abundant as a right GI-semigroup.

It is easy to write down laws that define the class of double GI-semigroups and double interior semigroups, using Proposition 4.8 and its dual, and adding the requirement that $I(S)=J(S)$.

Proposition 6.2

Let S be a semigroup. If S is a double GI-semigroup with respect to $E\subseteq E(S)$, with associated unary operations I, J then the following laws are satisfied:

• $I(x)x=I(x)$

• $I(x)y=I(x) \Rightarrow I(x)I(y)=I(x)$

• $I(x)I(y)=I(x) \& I(y)I(x)=I(y) \Rightarrow I(x)=I(y)$

• $xJ(x)=J(x)$

• $yJ(x)=J(x) \Rightarrow J(y)J(x)=I(x)$

• $J(x)J(y)=J(y) \& J(y)J(x)=J(x) \Rightarrow J(x)=J(y)$

• $I(J(x))=J(x), J(I(x))=I(x)$.

In this case, $I(S)=J(S)=E$, which is pre-reduced (that is, $\le _r$ and $\le _l$ are partial orders).

Conversely, if S is a biunary semigroup satisfying the above laws, then it is a double GI-semigroup with respect to $I(S)=J(S)$.

The double GI-semigroup is double interior if and only if it satisfies the laws

• $I(x)I(y)=I(x)\Rightarrow I(y)I(x)=I(x)$ and

• $I(x)I(y)=I(y)\Rightarrow I(y)I(x)=I(y)$.

The double GI-semigroup S is $FG^I$-abundant if and only if it satisfies the laws

• $xy=x \Rightarrow xI(y)=x$ and

• $xy=y \Rightarrow J(x)y=y$.

These laws can be re-phrased using Proposition 4.10, and then simplified somewhat, as follows:

Proposition 6.3

The laws for double GI-semigroups may be equivalently given as semigroups with unary operations I, J satisfying the following laws:

• $I(x)x=I(x)$

• $I(xI(yI(x)))=I(yI(x))$

• $I(x)y=I(x) \Rightarrow I(x)I(y)=I(x)$

• $xJ(x)=J(x)$

• $J(J(J(x)y)x)=J(J(x)y)$

• $yJ(x)=J(x) \Rightarrow J(y)J(x)=J(x)$

• $I(J(x))=J(x), J(I(x))=I(x)$.

Hence the laws for $FG^I$-abundant double GI-semigroups may be given as the laws above with the third and sixth replaced respectively by

• $xy=x \Rightarrow xI(y)=x$ and

• $xy=y \Rightarrow J(x)y=y$.

Proof

Mostly this follows from Proposition 4.10 and its dual. The redundancy of the law $I(I(s))=I(s)$ follows because we may argue using the above laws that for all $s\in S$, $I(I(s))=I(J(I(s)))=J(I(s))=I(s)$; similarly for $J(J(s))=J(s)$. $\square$

We note that PT(X) is an example of a double GI-semigroup, in which $I(S)=J(S)$ consists of all restrictions of the identity function on X: define I(f) to be the identity map on the fix-set of $f\in PT(X)$, and define J(f) to be the restriction of I(f) to those $x\in {{\,\textrm{dom}\,}}(I(f))$ whose inverse image set is just $\{x\}$. In this case I(S) is a semilattice and one may easily verify the claim that PT(X) is a double GI-semigroup by verifying the order-theoretic characterisations of I, J in terms of the natural order on I(S).

If each of the one-sided structures associated with a double GI-semigroup has large meets, then the laws in Proposition 6.3 may be restated equationally, courtesy of Proposition 5.6. However, in general we have the following.

Proposition 6.4

The class of double GI-semigroups is a proper quasivariety of biunary semigroups. So is the class of $FG^I$-abundant double GI-semigroups.

Proof

Let $X=\{w,x,y,z\}$ and let $S=\{e,f,k,l,a,1,0\}\subseteq PT(X)$ be such that

$$\begin{aligned}e=\{(w,w),(x,w),(z,w)\}, f=\{(w,z),(y,z),(z,z)\},\\k=\{(w,z),(x,z),(z,z)\}, l=\{(w,w),(y,w),(z,w)\},\end{aligned}$$

a is the permutation $\{(w,z),(x,y),(y,x),(z,w)\}$, 1 is the identity function on X and 0 is the empty function. Then S is a subsemigroup of PT(X) with the following multiplication table:

$$\begin{aligned} \begin{array}{c|ccccccc} \cdot &{}e&{}f&{}k&{}l&{}a&{}1&{}0\\ \hline e&{}e&{}k&{}k&{}e&{}k&{}e&{}0\\ f&{}l&{}f&{}f&{}l&{}l&{}f&{}0\\ k&{}e&{}k&{}k&{}e&{}e&{}k&{}0\\ l&{}l&{}f&{}f&{}l&{}f&{}l&{}0\\ a&{}l&{}k&{}f&{}e&{}1&{}a&{}0\\ 1&{}e&{}f&{}k&{}l&{}a&{}1&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0 \end{array} \end{aligned}$$

Note that

$$\begin{aligned}{{\,\textrm{fix}\,}}(e)=\{e,l,0\}={{\,\textrm{fix}\,}}(l), {{\,\textrm{fix}\,}}(f)=\{f,k,0\}={{\,\textrm{fix}\,}}(k), {{\,\textrm{fix}\,}}(a)=\{0\}={{\,\textrm{fix}\,}}(0),\end{aligned}$$

and ${{\,\textrm{fix}\,}}(1)=S$, so ${\mathcal F}$ partitions S into $\{e,l\},\{f,k\},\{a,0\}, \{1\}$. So if $E=\{e,f,1,0\}\subseteq E(S)$, then S is $(E,{\mathcal F})$-amiable, and so can be made into an $F^I$-abundant left GI-semigroup by setting $I(l)=e, I(k)=f, I(a)=0$, with all members of E fixed by I. Similarly,

$$\begin{aligned}{{\,\textrm{lfix}\,}}(e)=\{e,k,0\}={{\,\textrm{lfix}\,}}(k), {{\,\textrm{lfix}\,}}(f)=\{f,l,0\}={{\,\textrm{lfix}\,}}(l), {{\,\textrm{lfix}\,}}(a)=\{0\}={{\,\textrm{lfix}\,}}(0),\end{aligned}$$

and ${{\,\textrm{lfix}\,}}(1)=S$, so ${\mathcal G}$ partitions S into $\{e,k\},\{f,l\},\{a,0\}, \{1\}$. So S is $(E,{\mathcal G})$-amiable, and so can be made into a $G^J$-abundant right GI-semigroup by setting $J(k)=e, J(l)=f, J(a)=0$, with all members of E fixed by J. Of course, $I(S)=J(S)=E$ so S is an $FG^I$-abundant double GI-semigroup.

A congruence on S that respects I and J is obtained by grouping $\{e,f,k,l\}$ with all other elements kept separate. Just using e to stand for the four-element congruence class gives the following factor semigroup T.

$$\begin{aligned} \begin{array}{c|cccc} \cdot &{}e&{}a&{}1&{}0\\ \hline e&{}e&{}e&{}e&{}0\\ a&{}e&{}1&{}a&{}0\\ 1&{}e&{}a&{}1&{}0\\ 0&{}0&{}0&{}0&{}0 \end{array} \end{aligned}$$

Moreover $I(a)=J(a)=0$ with $I(x)=J(x)=x$ for all $x\in E=\{e,1,0\}$. Note that $ea=e$, but $eI(a)=0\ne e$, so T equipped with I is not even a left GI-semigroup. $\square$

In this last proof, S cannot have large meets with respect to at least one of its one-sided GI-semigroup structures, by the earlier comments, even though I(S) is evidently reduced, showing that the assumption that $I(S)=E(S)$ is necessary in Proposition 5.3.

7 Two-sided GI-semigroups

In [15], it was shown that every $({\mathcal F},{\mathcal G})$-amiable semigroup S (meaning one that is both ${\mathcal F}$-amiable and ${\mathcal G}$-amiable) is such that ${\mathcal F}={\mathcal G}$, which therefore equals ${\mathcal K}$, and so when viewed as an $FG^I$-abundant double GI-semigroup in which $I(S)=J(S)=E(S)$, the two induced unary operations are equal. This helps motivate the following.

Definition 7.1

Let S be a double GI-semigroup (resp. double interior semigroup). We say it is a two-sided GI-semigroup (resp. two-sided interior semigroup) if $I(s)=J(s)$ for all $s\in S$, and we say it is K-abundant if it is $FG^I$-abundant as a double GI-semigroup.

Examples of two-sided GI-semigroups include any central left GI-semigroup, since we may simply define $J(s)=I(s)$ for all s. Similarly, we have the following easy consequence of Proposition 3.13.

Corollary 7.2

Suppose S is a double GI-semigroup in which $I(S)=E(S)$. Then it is a two-sided GI-semigroup.

Not all two-sided GI-semigroups have $I(S)=E(S)$. For example, if $S=\{0,1\}$ with 0 a zero and 1 an identity element, then defining $I(s)=1$ for all $s\in S$ is easily seen to make the semilattice S into a two-sided GI-semigroup. Moreover, not every double GI-semigroup is a two-sided GI-semigroup, as the example of a double GI-semigroup given in the proof of Proposition 6.4 shows.

We noted earlier that every $({\mathcal F},{\mathcal G})$-amiable semigroup can be viewed as a K-abundant two-sided GI-semigroup. But the converse is true also: for if S is a K-abundant two-sided GI-semigroup, then $x^2=x$ implies $xI(x)=x$, yet $xI(x)=I(x)$, so $x=I(x)$, and so $I(S)=E(S)$, and so S is $({\mathcal F},{\mathcal G})$-amiable.

In contrast to Proposition 6.4 and the properties of the example given in its proof (it was not left reduced), for two-sided GI-semigroups we have the following.

Theorem 7.3

The class of two-sided GI-semigroups is a variety given by the following laws:

• 1. $I(x)x=I(x)=xI(x)$;

• 2. $I(I(x))=I(x)$;

• 3. $I(I(x)y)=I(yI(x))$;

• 4. $I(I(x)y)I(y)=I(I(x)y)$.

Moreover I(S) is reduced and has large meets, so I(ef) is the meet of any $e,f\in I(S)$ with respect to the natural order, and the classes of two-sided GI-semigroups and two-sided interior semigroups coincide.

Proof

Suppose S is a two-sided GI-semigroup. Then it is both a left GI-semigroup and a right GI-semigroup with respect to I, so the first two laws hold by Proposition 4.10. Now suppose $y\in S$ and $e\in I(S)$. Then we have that

$$\begin{aligned}I(eye)ye=I(eye)eyeye=I(eye)eyeeye=I(eye)eye=I(eye),\end{aligned}$$

so $I(eye)I(ye)=I(eye)$ as well. Moreover, $I(ye)eye=I(ye)yeeye=I(ye)yeye=I(ye)ye=I(ye)$, so $I(ye)I(eye)=I(ye)$. So $I(ye)\sim _r I(eye)$, and so by the right pre-reduced property, $I(ye)=I(eye)$. By the left/right symmetry of the laws in Proposition 6.3, we also have that $I(ey)=I(eye)$, and so $I(ey)=I(ye)$. Hence the third law holds, so by Proposition 5.6, S has large meets as a left GI-semigroup, I(S) is left reduced, and $I(I(ey)e)I(y)=I(I(ey)e)$ for all $e\in I(S)$ and $y\in S$. Hence, for all $y\in S$ and $e\in I(S)$, $I(ey)e=I(ye)e=I(ye)yee=I(ye)ye=I(ye)=I(ey)$, and so

$$\begin{aligned}I(ey)I(y)=I(I(ey))I(y)=I(I(ey)e)I(y)=I(I(ey)e)=I(I(ey))=I(ey),\end{aligned}$$

proving the final law above. Arguing dually, we see that S is also a right GI-semigroup with large meets with respect to I, and so is right reduced by the dual of Proposition 5.6. Hence I(S) is reduced, so both $\le _l$ and $\le _r$ are the natural order.

Conversely, assume the given laws hold for S. Then for all $e\in I(S)$ and $y\in S$, $I(ey)e=I(ye)e=I(ye)yee=I(ye)ye=I(ye)=I(ey)$, and so $I(I(ey)e)I(y)=I(I(ey))I(y)=I(ey)I(y)$. So by Proposition 5.6, S is a left GI-semigroup with respect to I. Indeed, we may evidently dualise this entire argument to obtain that S is a right GI-semigroup with respect to I providing only that the law $I(x)I(xI(y))=I(xI(y))$, dual to the final law in the theorem statement, holds. But we have that

$$\begin{aligned} I(xI(y))=I(I(xI(y)))=I(I(I(y)x))=I(I(I(y)x)I(x))=I(I(x)I(xI(y))),\end{aligned}$$

so

$$\begin{aligned}{} & {} I(xI(y))=I(x)I(xI(y))I(I(x)I(xI(y)))\\{} & {} =I(x)I(xI(y))I(xI(y))=I(x)I(xI(y)), \end{aligned}$$

as required. So S is a two-sided GI-semigroup with respect to I. $\square$

It follows that the class of two-sided GI-semigroups is a subvariety (of unary semigroups) of the class of left GI-semigroups with large meets.

There is an order-theoretic characterisation of two-sided GI-semigroups, following from Corollary 4.6 and its dual, that relates to the discussion of generalised interior operations given in Sect. 2.

Corollary 7.4

Suppose S is a semigroup with $E\subseteq E(S)$ such that for all $s\in S$, there is $I(s)\in E$ such that I(s) is both the largest $e\in E$ under the natural order such that $es=e$ and the largest $e\in E$ under the natural order such that $se=e$. Then S is a two-sided GI-semigroup with respect to I, and $I(S)=E$. Moreover, every two-sided GI-semigroup arises in this way.

If S is a two-sided GI-semigroup, although I(S) must be reduced, stronger properties need not hold.

Example 7.5

A K-abundant two-sided GI-semigroup in which I(S) does not commute and is not a band.

Let $X=\{x,y\}$ and let $S=\{e,f,0,a\}\subseteq PT(X)$ be such that

$$\begin{aligned}e=\{(x,y),(y,y)\}, f=\{(x,x)\}, 0=\emptyset , a=\{(x,y)\}.\end{aligned}$$

Then S is a subsemigroup of PT(X) with the following multiplication table:

$$\begin{aligned} \begin{array}{c|cccc} \cdot &{}e&{}f&{}0&{}a\\ \hline e&{}e&{}0&{}0&{}0\\ f&{}a&{}f&{}0&{}a\\ 0&{}0&{}0&{}0&{}0\\ a&{}a&{}0&{}0&{}0 \end{array} \end{aligned}$$

Now $E(S)=\{e,f,0\}$, and it is routine to verify that ${\mathcal F}={\mathcal G}={\mathcal K}$ partitions S into $\{e\},\{f\},\{0,a\}$, all of which contain (necessarily unique) idempotents. But $ef=0$ while $fe=a\not \in E(S)$.

This example also shows that the converse of Corollary 5.5 fails, even under additional assumptions: by Theorem 7.3, the above example has large meets (with respect to any of $\le _r,\le _l$ and $\le$, which are all equal) and I(S) is reduced, but I(S) does not commute (as $ef\ne fe$).

By contrast, Example 5.8, which is commutative, can be viewed as a K-abundant two-sided GI-semigroup; this shows that the class of K-abundant two-sided GI-semigroups is not a variety (even when I(S) is a semilattice), in contrast to the larger class of two-sided GI-semigroups.

8 Digression: $K_E$-semigroups

As noted in Proposition 3.12, for a ${\mathcal K}$-abundant (respectively, an $(E,{\mathcal K}_E)$-abundant) semigroup, amiability is automatic; for $s\in S$, define K(s) to be the unique $e\in E(S)$ (respectively, $e\in E$) for which $(s,e)\in {\mathcal K}$ (respectively, $(s,e)\in {\mathcal K}_E$). In particular, no special assumptions are required regarding K(S): it need be neither left nor right pre-reduced in general, so we do not expect to obtain left or right GI-semigroups.

Definition 8.1

We say the unary semigroup S with unary operation K is a $K_E$ -semigroup (resp. a K-semigroup) if S is $(E,{\mathcal K}_E)$-abundant (resp. ${\mathcal K}$-abundant) and K is defined as above.

It follows immediately that if one defines a $K_E$-semigroup from an $(E,{\mathcal K}_E)$-abundant semigroup S, then $K(S)=E$.

A two-sided GI-semigroup is nothing but a semigroup which is both $(E,{\mathcal F}_E)$-amiable and $(E,{\mathcal G}_E)$-amiable and such that ${\mathcal F}_E={\mathcal G}_E$, so it is trivially also $(E,{\mathcal K}_E)$-amiable as well. This means that the two-sided GI-semigroups just studied form a subclass of $K_E$-semigroups.

In fact laws for K-semigroups are given in Proposition 4.3 of [15] (again note the error in the first law given there). These laws are:

• 1. $K(x)x=K(x)=xK(x)$;

• 2. $xy=x$ implies $xK(y)=x$; and

• 3. $yx=x$ implies $K(y)x=x$.

A simple example was given in [15] to show that these laws are properly quasiequational. In the same way, we obtain the following.

Theorem 8.2

The class of $K_E$-semigroups is a proper quasivariety given by the following laws.

• 1. $K(x)x=K(x)=xK(x)$;

• 2. $K(x)y=K(x)$ implies $K(x)K(y)=K(x)$; and

• 3. $yK(x)=K(x)$ implies $K(y)K(x)=K(x)$.

Proof

Suppose S is an $(E,{\mathcal K}_E)$-abundant semigroup with K defined as above. The argument that the above laws all hold is very similar to the proof of the corresponding direction of Proposition 4.8: they essentially state that $(x,K(x))\in {\mathcal F}_E$ and $(x,K(x))\in {\mathcal G}_E$, that is, $(x,K(x))\in {\mathcal K}_E$ for all $x\in S$ (noting only that $E=\{K(s)\mid s\in S\}$).

Conversely, assume the above laws hold. Since $K(x)x=K(x)$, then by the second law, $K(x)K(x)=K(x)$, so $K(x)\in E(S)$ for all $x\in S$. Let $E=\{K(s)\mid s\in S\}$; then once again, the above laws simply spell out that S is $(E,{\mathcal K}_E)$-abundant.

Consider the left GI-semigroup S as in the proof of Theorem 4.8. We saw there that ${\mathcal F}_E$ partitions S as follows: $\{a,f\},\{e\},\{g\},\{1\}$. On the other hand, ${{\,\textrm{lfix}\,}}_E(a)=\{e,g,f\}={{\,\textrm{lfix}\,}}_E(e)={{\,\textrm{lfix}\,}}_E(g)={{\,\textrm{lfix}\,}}_E(f)$, while ${{\,\textrm{lfix}\,}}_E(1)=E$, and so ${\mathcal G}_E$ partitions S as follows: $\{a,e,g,f\},\{1\}$. So ${\mathcal F}_E\subseteq {\mathcal G}_E$ and so ${\mathcal K}_E={\mathcal F}_E$, so we have ${\mathcal K}_E$-abundance and so the left GI-semigroup structure is also a $K_E$-semigroup structure. Using the same congruence as used previously, the resulting quotient structure does not satisfy the second law in Proposition 8.2. So the quasivariety of $K_E$-semigroups is proper. $\square$

We have the following easy consequence of Proposition 3.9 and its dual.

Proposition 8.3

If S is ${\mathcal K}$-abundant then ${\mathcal K}={\mathcal K}_{E(S)}$, and so S is ${\mathcal K}_{E(S)}$-abundant; hence every K-semigroup is a $K_E$-semigroup for which $K(S)=E(S)$.