Regular semigroups which are subdirect products of a band and a semilattice of groups

Petrich, Mario

doi:10.1017/s0017089500001701

Cited by 22 publications

(10 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It thus follows from [13,Theorem 3.2] that (S; ·) is a subdirect product of a band B and a semilattice L of groups. By the definition of subdirect product, B and L are homomorphic images of S, so B and L satisfy every identity satisfied by S; hence B and L have the generalized entropic property.…”

Section: Theorem 68 Let (S; ·) ∈ S N Then (S; ·) Has the Generalizmentioning

confidence: 99%

Entropicity and generalized entropic property in idempotent $$n$$ n -semigroups

Lehtonen

Pilitowska

2015

Semigroup Forum

View full text Add to dashboard Cite

We investigate entropicity and the generalized entropic property in nsemigroups. These two properties are not equivalent for n-semigroups in general, but we show that there are certain classes of idempotent n-semigroups for which entropicity and the generalized entropic property are equivalent, such as Mal'cev n-semigroups and idempotent n-semigroups derived from binary semigroups. Moreover, we present an alternative description of entropicity in the variety of semigroups satisfying the identity x n ≈ x for some n ≥ 2. We also obtain a simple representation of the free ternary Mal'cev semigroup on two generators.

show abstract

Section: Theorem 68 Let (S; ·) ∈ S N Then (S; ·) Has the Generalizmentioning

confidence: 99%

Entropicity and generalized entropic property in idempotent $$n$$ n -semigroups

Lehtonen

Pilitowska

2015

Semigroup Forum

View full text Add to dashboard Cite

show abstract

“…Moreover, for any (/, yy) in E(e) x H e ; (/, yy)%(f, e) in P, the inverse of (/, yy) in H ifJ) is (/, y~*y). We refer the reader to [1] and [12] …”

Section: Punched Spined Productsmentioning

confidence: 99%

Left regular bands of groups of left quotients

El-Qallali

1991

Glasgow Math. J.

View full text Add to dashboard Cite

Introduction. In this paper we characterize semigroups 5 which have a semigroup Q of left quotients, where Q is an 9?-unipotent semigroup which is a band of groups. Recall that an 01-unipotent (or left inverse) semigroup S is one in which every 3?-class contains a unique idempotent. It is well-known that any 9?-unipotent semigroup 5 is a regular semigroup in which the set of idempotents is a left regular band in that efe = ef for any idempotents e, f in S. 3£-unipotent semigroups were studied by several authors, see for example [1] and [13]. Bailes [1] characterized 9?-unipotent semigroups which are bands of groups. This characterization extended the structure of inverse semigroups which are semilattices of groups. Recently, Gould studied in [7] the semigroup S which has a semigroup Q of left quotients where Q is an inverse semigroup which is a semilattice of groups.Many definitions of semigroups of quotients have been proposed and studied. For a survey, the reader may consult Weinert's paper [14]. These definitions have been motivated by corresponding definitions in ring theory. In this paper we are concerned with a concept of semigroups of left quotients adopted by Fountain and Petrich [5]. The definition proposed there is restricted to completely 0-simple semigroups of left quotients. The idea is that a completely 0-simple semigroup Q containing a subsemigroup S is a semigroup of left quotients of S if every element q in Q can be written as q = a~xb for some elements a, b in 5 with a 1 =t 0 and a" 1 the inverse of a in the group $?-class H a of Q. In this case 5 is also called a left order in Q. This definition and its dual were used in [5] to characterize semigroups 5 which have a completely 0-simple semigroup of quotients. An extension of this definition was used in [6] to obtain necessary and sufficient conditions for a semigroup 5 to have a bisimple inverse w-semigroup of left quotients. This extended definition was also used in [7] to characterize semigroups S which have a semigroup Q of left quotients where Q is an inverse semigroup which is a semilattice of groups. In this paper we consider the corresponding problem for 92-unipotent semigroups which are bands of groups.After preliminary results, we obtain in Section 2, the necessary and sufficient conditions for a semigroup 5 to have a semigroup Q of left quotients where Q is an !%-unipotent semigroup which is a band of groups. This result will be used in Section 3 together with the characterization of ^-unipotent semigroups which are bands of groups in terms of spined products to obtain an alternative structure for a semigroup 5 to have a left regular band of groups as a semigroup of left quotients. Section 4 is devoted to the case where the left orders are in a class of 9?*-unipotent semigroups.We use the notation and terminology of Howie [9]. Other undefined terms can be found in Fountain's paper [4].

show abstract

“…We will use the notation and terminology of Howie [2] and Petrich [7]. In addition to those in the introduction, we state a few frequently occurring symbols and concepts.…”

Section: Preliminariesmentioning

confidence: 99%

“…)°, rtrp = tr(p v^)°, where the join is taken in the lattice of equivalence relations on S and 8° means the greatest congruence on S contained in an equivalence relation 6 on S. The triple (ltrp, ker p, r tr p) uniquely determines the congruence p. By pK, pT, pT h pT r ; pk, pt, pt,, pt r we denote the greatest congruences on S having the same kernel, trace, left trace and right trace, respectively, as p; and the same with least replacing greatest. This provides eight operators on #(S) which we denote by K, T,.... Then K and k, T and t, T, and r,, T r and t r induce equivalence relations Jf, i 7 ", &l, F r , respectively, on #(S). The first one 180 M. PETRICH of these is a complete A-congruence, the remaining ones are complete congruences.…”

Section: Introductionmentioning

confidence: 99%

Congruence lattices of regular semigroups related to kernels and traces

Petrich¹

1991

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

The kernel-trace approach to congruences on a regular semigroup S can be refined by introducing the left and right traces. This induces eight operators on the lattice of congruences on S: t,, k, t r ; T h K, T r ; t, T. We describe the lattice of congruences on S generated by six 3-element subsets of the set {cot,, cok, a>t r , eT,, e.K, eT,} where co and E denote the universal and the equality relations. This is effected by means of a diagram and in terms of generators and relations on a free distributive lattice, or a homomorphic image thereof. We perform the same analysis for the lattice of congruences on S generated by the set {eK, cok, ET, cot}.

show abstract

Regular semigroups which are subdirect products of a band and a semilattice of groups

Cited by 22 publications

References 11 publications

Entropicity and generalized entropic property in idempotent $$n$$ n -semigroups

Entropicity and generalized entropic property in idempotent $$n$$ n -semigroups

Left regular bands of groups of left quotients

Congruence lattices of regular semigroups related to kernels and traces

Contact Info

Product

Resources

About