The natural partial order on an abundant semigroup

Abstract

In this paper we will study the properties of a natural partial order which may be defined on an arbitrary abundant semigroup: in the case of regular semigroups we recapture the order introduced by Nambooripad [24]. For abelian PP rings our order coincides with a relation introduced by Sussman [25], Abian [1,2] and further studied by Chacron [7]. Burmistrovic [6] investigated Sussman's order on separative semigroups. In the abundant case his order coincides with ours: some order theoretic properties of such semigroups may be found in a paper by Burgess [5].Many properties and constructions on abundant semigroups may be described in terms of its natural partial order: one of the main results of Section 2 is the connection we establish between idempotent connectedness and the partial order being, in some sense, self-dual. In Section 3 we extend Nambooripad's Theorem 3.3 [24] and show that the order is compatible with the multiplication on a concordant semigroup just when the semigroup is locally type A. In Section 4 we obtain a description of the finest 0-restricted primitive good congruence on a concordant semigroup. Section 1 is a preliminary section in which we consider, in particular, the behaviour of good homomorphisms between abundant semigroups and obtain a generalisation of Lallement's Lemma. PreliminariesWe will assume some familiarity with the contents of [11] and [15]: the only divergence from the terminology established there is that we prefer to call *-ideals good ideals, bringing them in line with the good homomorphisms.The class of idempotent connected (IC) abundant semigroups was introduced by El-Qallali and Fountain [11]. We begin this section with an alternative characterisation of these semigroups due to Fountain, which, for the purposes of this paper, may serve as an alternative definition.

Locally adequate semigroup algebras

“…The next result, which is due to Lawson [16], gives a characterization of the natural partial order on an abundant semigroup. Lemma 4.5.…”

“…An abundant semigroup S is called idempotent connected (IC) [17], if for all a 2 S , a 2 R a .S / \ E.S / and A semigroup S is said to satisfy the regularity condition [16] if for all idempotents e and f of S the element ef is regular. If this is the case, the sandwich set S.e; f / D fg 2 V .ef / \ E.S / j ge D fg D gg of idempotents e and f is non-empty, and takes the form…”

On fuzzy congruences of a type-B semigroup I

“…First, we state some known results and notations which will be frequently used throughout the paper. [9], Lawson define a natural partial order relation " ≤ " on an abundant semigroup S as follows: (∀a, b ∈ S) a ≤ b if and only if there are idempotents e, f ∈ S, such that a = eb = bf.…”

“…(B2) if for all a ∈ S, e ∈ E(S), e ≤ a * , then there is an element f ∈ E(S 1 ) such that e = (f a) * , where " ≤ " is a natural partial order on E(S) ( i.e., (∀e, f ∈ E(S)) e ≤ f ⇔ e = ef = f e (see, [9])).…”

A natural partial order for semigroups

“…There have been numerous attempts to extend this ordering from Es to all of S defining it by means of the multiplication of S and postulating that its restriction to Es coincides with the partial order (1) (see Burgess and Raphael [2], Hartwig [4], Lawson [8], McAlister [9] and Nambooripad [10]). The reason for the interest in such natural orders lies in the obvious fact that such an ordering can provide additional information on a given semigroup since it reflects its multiplication in a particular way.…”