A semigroup S is called E-inversive if for every a ∈ S there exists x ∈ S such that ax is idempotent. S is called E-semigroup if the set of idempotents of S forms a subsemigroup. In this paper some special congruences on E-inversive E-semigroups are investigated, such as the least group congruence, a certain semilattice congruence, some regular congruences and a certain idempotent-separating congruence.

Certain congruences on E-inversive E-semigroups

We show that there is an inclusion-preserving bijection between the set of all normal subsemigroups of a semigroup S and the set of all group congruences on S. We describe also group congruences on E-inversive (E-)semigroups. In particular, we generalize the result of Meakin (J. Aust. Math. Soc. 13:259–266, 1972) concerning the description of the least group congruence on an orthodox semigroup, the result of Howie (Proc. Edinb. Math. Soc. 14:71–79, 1964) concerning the description of ρ∨σ in an inverse semigroup S, where ρ is a congruence and σ is the least group congruence on S, some results of Jones (Semigroup Forum 30:1–16, 1984) and some results contained in the book of Petrich (Inverse Semigroups, 1984). Also, one of the main aims of this paper is to study of group congruences on E-unitary semigroups. In particular, we prove that in any E-inversive semigroup, $\mathcal{H}\cap\sigma\subseteq\kappa$, where κ is the least E-unitary congruence. This result is equivalent to the statement that in an arbitrary E-unitary E-inversive semigroup S, $\mathcal{H}\cap\sigma= 1_{S}$.

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Congruences and group congruences on a semigroup

A semigroup S is called E-inversive if for any a∈S there exists x∈S such that ax∈E(S). In this paper, the regular congruences on an E-inversive semigroup are investigated. It is proved that each regular congruence ρ on an E-inversive semigroup S is uniquely determined by the pair (ker ρ,tr ρ) and an abstract characterization of regular congruence ρ by means of the pair (ker ρ,tr ρ) (called regular congruence pair) is given. Also it is shown that each regular congruence on S is uniquely determined by its kernel normal system and a description of the regular congruences on S in terms of their kernel normal systems is given.

Regular congruences on an E-inversive semigroup

For an ordered semigroup (S, ·,≤), it is natural to ask the following questions: Is the quotient semigroup (S/ρ, ·)( where “.” is the usual multiplication on S/ρ) an ordered semigroup for a congruence ρ on S? If there exists an ordered relation “ ” on S/ρ such that the semigroup (S/ρ, ·) is an ordered semigroup (S, ·, ) , then whether the natural homomorphism from S onto S/ρ is isotone or not? In [11] Blyth and Janowtz defined the residuated semigroups and Dubreil equivalences, in [13] Fuchs gave a Dubreil-Jacotin congruence θ on a residuated ordered semigroup S with an identity e such that S/θ is an ordered group and also the natural homomorphism from S onto S/θ is isotone. Later in [14], for a commutative residuated lattice-ordered semigroup S with an identity e being the greatest element in S, McCorthy introduced a congruence θ characterized by some subsemigroup M of S such that S/θ is an ordered semigroup, and he proved that if T is a residuated ordered semigroup and there is a homomorphism h from S onto T which preserves residuals, then there is an isomorphism f from S/θ onto T such that fθ = h. Recently, Kehayopulu and Tsingelis (see [1], [2]) introduced the concept of pseudoorder on an ordered semigroup S and proved that if σ is a pseudoorder on S, then there exists a congruence σ on S such that S/σ is an ordered semigroup. In the same paper a necessary and sufficient condition such that S is a subdirect product of some ordered semigroups was given and two isomorphism theorems of S were established. In [7] we introduced the concept of regular semilattice congruence on an ordered semigroup S , and proved that semilattice congruence N on S is the least regular semilattice congruence on S. Earlier than [7] Kehayopulu and Tsingles in [15] proved the same result as in [7]. As a continuation of the investigation in [1], [2] and [7], we study in detail a congruence ρ on an ordered semigroup S such that S/ρ is an ordered semigroup and also the natural homomorphism from the semigroup S onto the semigroup S/ρ is isotone. The congruence with above property is called a regular congruence. We will prove that any Rees congruence ρI determined by an ideal I of S is a regular congruence (Proposition 2.7). Furthermore, we will establish a bijection mapping θ from the set of all ideals of S containing I to the ————————————————————— ∗ Supported by the Natinal Natural Science Foundation of China and Guangdong Provincial Natural Science Foundation of China.

On Regular, Strongly Regular Congruences on Ordered Semigroups

Congruences on an ample semigroup S are investigated. For any admissible congruence $$ \rho $$ on S, the minimum $$\sigma _{\rho }$$ (the maximum $$ \mu _{\rho }$$) admissible congruence on S whose restriction to E(S) is the trace of $$\rho $$ is obtained. An expression for a congruence which is contained in (contains) any admissible congruence of the same kernel is given. The concept of congruence pairs for S is introduced. It is shown that for any admissible congruence $$\rho $$ of S; ($${{\,\mathrm{tr}\,}}\rho , \ker \rho $$) is a congruence pair for S. Conversely, the congruence of S which contains (is contained in) any admissible congruence associated with a given congruence pair for S is formulated.

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Congruences on ample semigroups

Semigroups and their lattice of congruences II

Generalizing a property of regular resp. finite semigroups a semigroup S is called E-(0-) inversive if for every a ∈ S4(a ≠ 0) there exists x ∈ S such that ax (≠ 0) is an idempotent. Several characterizations are given allowing to identify the (completely, resp. eventually) regular semigroups in this class. The case that for every a ∈ S4(≠ 0) there exist x,y ∈ S such that ax = ya(≠ 0) is an idempotent, is dealt with also. Ideal extensions of E- (0-)inversive semigroups are studied discribing in particular retract extensions of completely simple semigroups. The structure of E- (0-)inversive semigroups satisfying different cancellativity conditions is elucidated. 1991 AMS classification number: 20M10.

Basic properties of e-inversive semigroups

Let S be a commutative semigroup. We consider the semigroup P(S) with respect to composition of all transformations p: S → S of the form x → a,x → xn or x → axn (a∈ S; n∈ N) and the semigroup P(S) containing only elements of the last two forms. Since all polynomials over S have the form a, xn or ax these transformations are the so-called polynomial functions over S. We investigate the relationship between the structures of S and$\bar P$(S) resp. P(S) — a criterion on the commutativity of S has been shown by means of polynomial functions in 2.

Polynomial functions over commutative semi-groups

A semigroup S (with zero) is called right resp. left (0-) quasiresiduated with respect to its natural partial order ≤S if for any a,b∈S (a,b≠0) there exists x∈S (x≠0) resp. y∈S (y≠0) such that ax≤Sb resp. ya≤Sb. It is shown that the most important semigroups of mappings are—at least for finite sets and finite-dimensional vector spaces—left and/or right (0-) quasiresiduated.

A property of transformation semigroups

A semigroup (S, ·) is called right (left) quasiresiduated if for any a, b in S there exists x in S such that ax ≤S b (xa ≤S b) with respect to the natural partial order ≤S of S. This concept has its origin in the theory of residuated semigroups, but can also be seen as a generalization of the right (left) simplicity of semigroups. It is first studied for totally-, resp., trivially-ordered semigroups, and then for semigroups with idempotents. In particular, the cases when (S, ≤S) is directed downwards and when S contains a zero (with respect to a more restrictive definition) are dealt with. Throughout, examples are given; in total, 30 classes of (often well-known) semigroups of this kind are specified.

Heinz Mitsch

Papers

Semigroups and their lattice of congruences II

Basic properties of e-inversive semigroups

Polynomial functions over commutative semi-groups

A property of transformation semigroups

Quasiresiduals in Semigroups with Natural Partial Order