Free right type A semigroups

Abstract

Introduction. The relation i£* is defined on a semigroup 5 by the rule that a!£*b if and only if the elements a, b of S are related by Green's relation =2" in some oversemigroup of 5. A semigroup S is an E-semigroup if its set £(5) of idempotents is a subsemilattice of S. A right adequate semigroup is an £-semigroup in which every i?*-class contains an idempotent. It is easy to see that, in fact, each i?*-class of a right adequate semigroup contains a unique idempotent [8]. We denote the idempotent in the if-class of a by a*. Then we may regard a right adequate semigroup as an algebra with a binary operation of multiplication and a unary operation *. We will refer to such algebras as "-semigroups. In [10], it is observed that viewed in this way the class of right adequate semigroups is a quasi-variety.In this paper, which is the promised sequel to [10], we are concerned with right type A semigroups. These are semigroups which are right adequate and in which ea = a(ea)* for each a e S and e e E(S); they form a sub-quasi-variety of the quasi-variety of all right adequate semigroups. Thus, from general results in universal algebra, we know that free right type A semigroups exist. It is the purpose of this paper to give an explicit description of these free objects and to discuss some of their properties.Our approach is to make use of the construction of free right h-adequate semigroups in [10], a right adequate semigroup being right h-adequate when the mapping a a : E(Sy -> E(Sy defined by xa a = (xa)* is a homomorphism for each element a of 5. By a "-congruence on a right adequate semigroup 5, we mean a congruence on S regarded as a "-semigroup, that is, a semigroup congruence p on 5 which also satisfies apb implies a*pb*. A "-congruence p on a right adequate semigroup S is called a right type A congruence if Sip is a right type A semigroup, where the semigroup Sip is made into a "-semigroup by defining (ap)* to be a*p. On any right adequate semigroup, there is a minimum right type A congruence and if y is this congruence on P x , the free right h-adequate semigroup on X, then P x ly is the free right type A semigroup on X. For any non-empty set X, we construct a semigroup A x isomorphic to P x ly which is analogous to Scheiblich's construction [20] of the free inverse semigroup on X.This construction allows us to obtain several results which are analogues of theorems on inverse semigroups. For example, the fact that free inverse semigroups are E-unitary gives rise to one proof that every inverse semigroup has an £-unitary cover [18, Theorem VIII. 1.10]. The corresponding result for right type A semigroups is that every right type A semigroup has a proper cover [8]. On a right type A semigroup, the minimum left cancellative congruence is denoted by a and the semigroup is proper if a n S£* = t. It is easily seen that ^4^ is proper and we give a new proof of the covering result modelled on that in the inverse case.After recalling the basic properties of right type A semigroups in Section 1, we devote Section 2 to th...

Free Adequate Semigroups

“…If X is a representative Σ-tree for X ∈ UT 1 (Σ) then X (+) is the isomorphism type of [9] Free adequate semigroups 373 the idempotent tree with the same underlying graph and start vertex as X , but with end vertex the start vertex of X . Dually, X ( * ) is the isomorphism type of the idempotent tree with the same underlying graph and end vertex as X , but with start vertex the end vertex of X .…”

Identities in the Algebra of Partial Maps

“…We direct the reader to [11] for further details on RC-semigroups and their relatives, but of particular interest are weakly right ample semigroups (see [7] for the left-sided version), and right type-A semigroups (see [5] for example). These closely related classes have seen reasonably extensive research.…”

“…The RC-semigroup reducts of members of TC generate the variety of twisted RCsemigroups with central closed elements. This variety was examined by Fountain [5] who showed that (after translation from right type-A semigroups to the language of RC-semigroups) it has decidable equational theory. Theorem 6.6 gives us the following contrasting result.…”

“…Twisted RC-semigroups also coincide with the type SL2 γ-semigroups of Batbedat [1], and are the variety generated by the right type-A semigroups studied by Fountain [5]. In [1] and [5], the free algebras in this class are constructed and the equational problem for the variety is solved. The first main result in the current paper is a positive solution to the corresponding problem for the variety of twisted agreeable semigroups.…”

“…The first main result in the current paper is a positive solution to the corresponding problem for the variety of twisted agreeable semigroups. The algorithm substantially extends that of [1] and [5] and turns out to be related to a weakened uniform word problem for semigroups; specifically to testing membership in finitely generated left congruences on free semigroups. As a corollary, we obtain an algorithm describing the equational theory of semigroups of partial maps with the operations of composition and relational intersection-these were shown to form a variety by Garvac'kiȋ [6].…”