Classification of some τ-congruence-free completely regular semigroups

Abstract

Let τ be an equivalence relation on a semigroup. We introduce τ -congruence-free semigroups, extending the notion of congruence-free semigroups, and classify all completely regular semigroups which are τ -congruence-free, where τ is one of Green's relations H, L and D respectively. Taking τ as H as well as D, this settles two open problems posed by M. Petrich and N.R. Reilly.

Relatively congruence-free regular semigroups

“…In Section 4 we completely determine the -CF, L-CF and H-CF completely semisimple semigroups. These results then easily specialize to those of [1] for completely regular semigroups. Apart from the generalization Lemma 2.1 of their first key result, our methods are independent of theirs.…”

“…There is therefore a sequence a = w0 → w1 → ⋯→ wn = b, where for each i, {wi−1,wi}={si(ga)ti, si(gb)ti}, si,ti∈S 1 and, for each i, wi ∈ J, ≥ J and ti ∈ J.…”

“…With u,v as stated, there is a sequence u = z0→z1→⋯→zn=v, where for each i, {zi−1,zi}={xiayi,xibyi}, xi,yi∈S 1 . Then for each i, zi ∈ J, ≥J and J, ≥J.…”

“…First recall [2, §1.6] that the congruence generated by a pair (a,b) of elements in a semigroup S is described as follows: distinct elements x,y are related if and only if there is a sequence of 'elementary transitions' x=z0→z1→⋯→zn=y of distinct elements of S; that is, for each i, {zi−1,zi}={siati,sibti}, for some si,ti∈S 1 .…”

“…Let S be a semigroup and τ any equivalence relation on S. Yu et al [1] defined S to be τ-congruence-free if any congruence ρ on S is either disjoint from τ or contains τ; following their terminology, we abbreviate the term τ-congruence-free to τ-CF. A congruence-free semigroup is then just an ω-CF semigroup, where ω is the universal relation, so this concept generalizes that of congruence-freedom.…”