1 Preliminaries

Let S be a semigroup and a∈S. An element x∈S is called a weak inverse of a if xax=x; the set of all weak inverses of a is denoted by W S (a). A semigroup S is said to be E-inversive if for every a∈S there is x∈S such that ax∈E S , where E S (or briefly E) is the set of idempotents of S (more generally, if A⊆S, then E A denotes the set of idempotents of A). If A⊆S, then by A ∗ we shall mean the set of all non-zero elements of A. Since each semigroup with zero is E-inversive, then we define a semigroup S with zero to be E-inversive if for all a∈S ∗ there exists x∈S such that \(ax \in E_{S}^{*}\). Finally, put \(W_{S}^{*}(a) = W_{S}(a) \setminus\{0\}\) (a∈S). Recall from [3] that a semigroup S [with zero] is E [∗]-inversive if and only if \(W_{S} ^{[*]}(a) \neq\emptyset\) for every a∈S [∗].

Lemma 1.1

A semigroup S [with zero] is E [∗]-inversive if and only if every [non-zero] ideal of S contains some [non-zero] idempotent of S.

Proof

Suppose that every non-zero ideal of S contains some non-zero idempotent of S, a∈S ∗. Then S 1 aS 1 contains at least one non-zero idempotent of S, that is, xay=e for some x,y∈S 1 (in fact, we may suppose that x,y∈S), \(e \in E_{S}^{*}\). Hence exaye=e, so (yex)a(yex)=yex≠0; otherwise 0=xa(yex)=(xay)ex=ex. Thus 0=exay=e, a contradiction. Consequently, \(yex \in W_{S} ^{*}(a)\).

The converse implication is clear. □

Lemma 1.2

Let S be an E [∗]-inversive semigroup. Then eSe is E [∗]-inversive for every \(e \in E_{S}^{[*]}\).

Proof

Observe first that e∈eSe, so eSe≠{0}. Let a∈(eSe)∗ and \(x\in W_{S}^{*}(a)\). Then x=xax=x(eae)x. Hence exe=(exe)a(exe). Furthermore, if exe=0, then we get xe=[(xe)a(ex)]e=(xea)(exe)=0, so x=(xe)a(ex)=0, a contradiction. Thus \(exe\in W_{eSe} ^{*}(a)\), as exactly required. □

We say that a semigroup S is a semilattice if a 2=a, ab=ba for all a,b∈S. Further, a congruence ρ on a semigroup S is called a semilattice congruence if S/ρ is a semilattice. It is clear that the least semilattice congruence η on an arbitrary semigroup exists. Finally, a semigroup is said to be η-simple if η=S×S.

The next lemma follows immediately from the Second Isomorphism Theorem.

Lemma 1.3

A homomorphic image of an η-simple semigroup is η-simple.

Let S be a semigroup. Recall that the natural partial order is the relation ≤, defined on E S by e≤f if e=ef=fe. We say that a semigroup S (without zero) has a least idempotent e if e≤f for every f∈E S . Note that If S has a zero, say 0, then clearly 0 is the least element of E S with respect to ≤, but in such a case, we may say that S has a least non-zero idempotent if \(E_{S}^{*}\) contains the least element with respect to the natural partial order.

Let A be an ideal of a semigroup S. We say that S is an ideal extension of the semigroup A by the semigroup T if the Rees semigroup S/A is isomorphic to T. Finally, an ideal P of a semigroup S is called prime if the condition ab∈P implies that a∈P or b∈P for all a,b∈S.

2 The main results

Remark that by Corollary 3.9 of [4], a semigroup S is η-simple if and only if S has no proper prime ideals.

Proposition 2.1

Let S≠S 0 be an η-simple semigroup with a least idempotent. Then S is E-inversive. Moreover, S is an ideal extension of a group by an η-simple semigroup.

Proof

Let e be the least element of E S . Then every ideal of S must contain e. Indeed, suppose by way of contradiction that there is an ideal A of S such that e∉A. Let B be the set theoretic union of all such ideals A of S. Then clearly B is the largest ideal of S such that e∉B. Next, consider the Rees quotient S/B. Notice that we may think about S/B as a semigroup with zero, where all products not falling in S/B are zero. Consider now an arbitrary non-zero ideal C of S/B. Then by construction of B, {e} must belong to C. Hence the intersection of all non-zero ideals of S/B contains {e}. In particular, S/B is E ∗-inversive (see Lemma 1.1). Also, B is a prime ideal of S. Indeed, let a,b∉B be such that ab∈B. Then fg∈B for some f,g∈E S ∖B (because S/B is E ∗-inversive). Hence e=efg∈B (which is a contradiction). It follows that S has a proper semilattice congruence (by the above remark), a contradiction with the assumption of the theorem. Consequently, every ideal of S must contain e. Thus S has a kernel G (say) and S is E-inversive (Lemma 1.1). Hence for every a∈S there exists x∈S such that ax,xa∈E S . Therefore e=(ax)e=a(xe)∈aS. We may equally well show that e∈Sa. It follows easily that S contains both a minimum left ideal L (say) and a minimum right ideal R (say). Furthermore, for every a∈S, La is a minimal left ideal of S (see [1], Lemma 2.32). Hence La=L, so L is an ideal of S (and L=L 2). We can show in a similar way that R is an ideal of S, so L=R=G=eS=Se (because Se⊆L,eS⊆R, since e∈L,R). Consequently, G=eSe. Indeed, evidently eSe⊂SeS=G. Also, G=GG=eSSe⊂eSe. By Lemma 1.2, G is an E-inversive monoid (with an identity element e). Moreover, if f∈E eSe , then fe=ef=f i.e. f≤e. Thus f=e. Consequently, G is a group ideal of S and so S is an ideal extension of the group G by the semigroup S/G which is η-simple, by Lemma 1.3, as required. □

Lemma 2.2

Let S≠S 0 be an η-simple semigroup with the least idempotent e. Then ea=ae for all a∈S.

Proof

Let a∈S. Then ea, ae∈eSe=eS=Se, where eSe is a group ideal of S (see the proof of Proposition 2.1). Hence e⋅ae=ae, ea⋅e=ea. Thus ea=ae. □

A congruence on a semigroup is called a group congruence if the quotient semigroup is a group.

Corollary 2.3

Let S≠S 0 be an η-simple semigroup with a least idempotent, say e. Then the mapping s→es of S onto the group eS is an epimorphism leaving the elements of eS fixed. Moreover, the congruence σ induced by this morphism, that is σ={(a,b)∈S×S:ea=eb}, is the least group congruence on S.

Proof

The first part of the result follows from Proposition 2.1 and Lemma 2.2. Further, if ρ is a group congruence on S, then clearly (s,es)∈ρ for every s∈S. Hence σ⊆ρ. □

Remark 1

Notice that if a semigroup S≠S 0 with the least idempotent e is η-simple, then ρ eS ∩σ=1 S and so S is a subdirect product of an (E-inversive) η-simple semigroup S/eS (with zero) and the group eS.

Further, the converse of Proposition 2.1 is valid.

Theorem 2.4

A semigroup S without zero is η-simple and has a least idempotent if and only if it is an (E-inversive) ideal extension of a group by an η-simple semi-group.

Proof

The direct part follows from Proposition 2.1.

Conversely, let G be a group ideal of S (with an identity e) and a∈S. Then ea∈G, say ea=g. It follows that g −1 ea=e∈Sa. We may equally well show that e∈aS, so e is the least idempotent of E S . Further, if ρ is a semilattice congruence on S, then (according to the proof of Theorem 5 in [5]) ρ∩(G×G)=G×G. It follows that ρ G ⊂ρ, where ρ G is the Rees congruence on S modulo G. Hence there is an epimorphism of S/ρ G onto S/ρ. In fact, this morphism induced on S/ρ G a semilattice congruence. Since S/ρ G is η-simple, then S/ρ must be trivial. Consequently, ρ=S×S, as exactly required. □

Remark that if a semigroup S is a left [right] group (i.e. \(S \times S = \mathcal{L}\) [\(\mathcal{R}\)]), then S is η-simple. Indeed, let S be a left [right] group. Then \(S \times S = \mathcal{L}\) [\(\mathcal{R}\)] \(\subseteq\mathcal{J} \subseteq\eta\).

Theorem 2.5

A semigroup S without zero is η-simple and has an idempotent e such that ef=e [fe=e] if and only if it is an (E-inversive) ideal extension of a left [right] group by an η-simple semigroup.

Proof

(⟹). Let ef=e for every f∈E S . We may equally well show like above (see the proof of Proposition 2.1) that e belongs to every ideal of S. Hence S has a kernel, say K. In particular, S is E-inversive. It follows that e∈Sa for every a∈S. Thus S contains a minimum left ideal L and L=La for all a∈S (so L=L 2). Therefore K=Se is a left simple semigroup (by Theorem 2.35 [1]) and so K is a left group (by the dual of Theorem 1.27 [1]). Consequently, S is an ideal extension of the left group K by the semigroup S/K which is η-simple.

(⟸). Let K be a left group ideal of S, e∈E K and a∈S. Then ea∈K, say ea=k. It follows that ek −1 ea=ek −1 k=e∈Sa, where k −1 is some inverse of k in K (since E K is a left zero semigroup). Hence if f∈E S , then e=sf for some s∈S. Thus ef=e. We have just shown that ef=e for all e∈E K , f∈E S . Further, if ρ is a semilattice congruence on S, then ρ∩(K×K)=K×K (by the preceding remark) and so ρ=S×S (by the proof of Theorem 2.4). □

Corollary 2.6

Let S be a simple semigroup. If S has an idempotent e such that ef=e [fe=e] for every f∈E S , then S is a left [right] group.

Proof

Indeed, in such a case, \(\mathcal{J} = S \times S\). It is almost evident (and also well-known) that \(\mathcal{J} \subseteq\eta\). Hence S is η-simple, so S contains a left [right] group ideal K. Thus S=K. □

Notice that if S is a completely simple semigroup (see [2], Sect. 3.2), then the Green’s relation \(\mathcal{H}\) is a band congruence on S (see Lemma III.2.4 in [2]). Further, every left [right] group S is completely simple and E S is a left [right] zero semigroup. It follows, from the above, that if S is a left [right] group, then \(S/\mathcal{H}\) is a left [right] zero semigroup.

A semigroup S is said to be congruence-free if it has exactly two congruences.

Proposition 2.7

Let S be a congruence-free semigroup without zero. If S has an idempotent e such that ef=e [fe=e] for every f∈E S , then S is a simple group.

Proof

Let ef=e for every f∈E S . Since S is congruence-free, then either η is the identity or the universal relation on S. In the former case, S is a semilattice, but then e is the zero of S, a contradiction with the assumption of the proposition. It follows that S is η-simple. By Theorem 2.5, S contains a left group ideal K. Hence S is itself a left group. From the above remark we conclude that either \(\mathcal{H} = 1_{S}\) or \(\mathcal{H} = S \times S\). In the former case, S must be a left zero semigroup. Since |S|>1, then the partition {{e},S∖{e}} of S induced a proper congruence on S, a contradiction. Thus \(\mathcal{H} = S \times S\), so E S ={e}, since \(\mathcal{H}\) separates idempotents of S. Consequently, S is a simple group. □

Next, consider a semigroup S with zero such that the set \(E_{S}^{*}\) contains a least idempotent, say e. Remark that fg≠0 for all \(f, g \in E_{S}^{*}\) (in fact, if \(e \in E_{S}^{*}\) has the property that ef=e [fe=e] for every \(f \in E_{S}^{*}\), then also gh≠0 for all \(g, h \in E_{S}^{*}\)).

Since a semigroup with zero adjoined has a proper semilattice congruence, then we shall say that a semigroup with zero is η ∗-simple if S has at most two semilattice congruences, namely: (i) S×S or (ii) the congruence induced by the partition {{0},S ∗}. Clearly, the partition {{0},S ∗} of a semigroup S with zero induces a semilattice congruence on S if and only if S is a semigroup with zero adjoined.

Recall that a semigroup S with zero is called a 0-group if S ∗ is a group.

Theorem 2.8

A semigroup S with zero is η ∗-simple and has a least non-zero idempotent if and only if it is an E ∗-inversive semigroup with zero adjoined (and so S ∗ is an E-inversive semigroup with a least idempotent) and it is an ideal extension of a 0-group by an η-simple semigroup.

Proof

(⟹). Let e be a least non-zero idempotent of S. We can show that every non-zero ideal of S contains e (see the proof of Proposition 2.1 and the above remark). In particular, S is E ∗-inversive (Lemma 1.1). Hence for every a∈S ∗ there is x such that xa is a non-zero idempotent of S. Thus e∈Sa. We may equally well show that e∈aS. Next, if a,b∈S ∗, then (by the above) e=xa,e=by for some x,y∈S. Hence e=x(ab)y and so ab∈S ∗. Consequently, S has no proper zero divisors. Thus S ∗ is an E-inversive semigroup with a least idempotent e and so S ∗ is an ideal extension of a group G by an η-simple semigroup (Theorem 2.4). It follows that S is an ideal extension of a 0-group G 0 by an η-simple semigroup. Indeed, S/G 0 must have a proper zero divisor (otherwise G 0 is a non-zero prime ideal of S).

The opposite implication follows easily from the proof of Theorem 2.4. □

A non-zero [left [right]] ideal A of a semigroup S with zero is called 0-minimum if it is contained in every non-zero [left [right]] ideal of S.

Further, a semigroup S with zero is called categorical if abc=0 implies that either ab=0 or bc=0 for all a,b,c∈S.

Finally, we have the following theorem.

Theorem 2.9

Let S be a categorical semigroup (with zero). Then S is η ∗-simple and has a non-zero idempotent e such that ef=e [fe=e] for every \(f \in E_{S}^{*}\) if and only if it is an E ∗-inversive semigroup with zero adjoined (and so S ∗ is an E-inversive semigroup with a least idempotent) and it is an ideal extension of a left [right] group with zero adjoined by an η-simple semigroup.

Proof

(⟹). Let ef=e for every \(f \in E_{S}^{*}\). We may equally well show like above that e belongs to every non-zero ideal of S. Hence S has a 0-minimum ideal K. In particular, S is E ∗-inversive. It follows that e∈Sa for all a∈S ∗. Thus S contains a 0-minimum left ideal L (and L=Le, so L=L 2). Therefore K=Se is a left 0-simple semigroup (by Theorem 2.35 in [1]), so K ∗ is a left simple semigroup (Theorem 2.27 in [1]). Thus K ∗ is a left group (by the dual of Theorem 1.27 in [1]). Further, suppose that ea=0 for some a∈S and let b∈S ∗. Then e=sb for some s∈S. Hence sba=0. Thus ba=0 (since S is categorical), so {0,a} is a left ideal of S. It follows that either {0,a}=K or a=0. Consequently, ea≠0 for all a∈S ∗. Therefore ab≠0 for all a,b∈S ∗. Indeed, if ab=0 for some a,b∈S ∗, then eb=0, a contradiction from the above. We conclude that S ∗ is an E-inversive semigroup, so S ∗ is an ideal extension of the left group K ∗ by the semigroup S/K which is η-simple (Theorem 2.5). Hence S is an ideal extension of the left group K with zero adjoined by the semigroup S/K which is η-simple, since K is not a prime ideal of S.

The opposite implication follows from the proof of Theorem 2.5. □