Adequate Semigroups

Abstract

A monoid in which every principal right ideal is projective is called a right PP monoid. Special classes of such monoids have been investigated in (2), (3), (4) and (8).There is a well-known internal characterisation of right PP monoids using the relation X* which is defined as follows. On a semigroup S, (a,b)E.Z£* if and only if the elements a,b of S are related by Green's relation 2? in some oversemigroup of S. Then a monoid S is a right PP monoid if and only if each i£*-class of S contains an idempotent. The existence of an identity element is not relevant for the internal characterisation and in this paper we study some classes of semigroups whose idempotents commute and in which each 2?*-class contains an idempotent. We call such a semigroup a right adequate semigroup since it contains a sufficient supply of suitable idempotents. Dually we may define the relation 91* on a semigroup and the notion of a left adequate semigroup. A semigroup which is both left and right adequate will be called an adequate semigroup.Any inverse semigroup is adequate and in investigating adequate semigroups it is natural to look for analogues of results for inverse semigroups. In (6), Howie determines the greatest idempotent separating congruence n on an inverse semigroup S and shows that Slfi is isomorphic to the semilattice of idempotents of S if and only if the idempotents in S are central. In an adequate semigroup there need not be a greatest idempotent separating congruence. However, on an inverse semigroup, /x is the largest congruence contained in 7R,. Defining Stf* to be X* D91*, we determine, in Section 2, the largest congruence ix L contained in X* on a right adequate semigroup and hence find the largest congruence fi contained in "3t* on an adequate semigroup. We also find that the analogue of the second result of Howie mentioned above holds.It is well known (see (10) or (7, Chapter V)) that if S is an inverse semigroup with semilattice of idempotents E, then there is an idempotent separating homomorphism 0 from S into the Munn semigroup T E such that 0 ° <T* = /x. and 6 restricts to an isomorphism between E and the semilattice of idempotents of T E . In Section 3, we define a class of right adequate semigroups which we call right type B semigroups. For any semilattice E we define a semigroup B(E) which is right type B and whose semilattice of idempotents E is isomorphic to E. For any right type B semigroup 5 with semilattice of idempotents E, there is a homomorphism 6:S-+B(E) such that $ o $~l = fx. L and 6 restricts to an isomorphism between E and E. In Section 4 we consider adequate semigroups and using T E , show that there is an analogue of Munn's result for a class of adequate semigroups which we call type A semigroups. 113

The natural partial order on an abundant semigroup

“…In fact each local submonoid of S is a semilattice, so S is an adequate locally inverse semigroup. If S were type A then for each idempotent e in S and each element a in S we would have eS 1 n aS 1 = eaS 1 -see Proposition 1.5 of [14]. Now, fa = b and fS 1 = {fz,b} and aS l = {a,z) and so fS l r>aS l = {z).…”

Retracts of trees and free left adequate semigroups

“…We also recall the definitions and basic properties of left, right and two-sided adequate semigroups, more details of which can be found in [5].…”

“…Adequate semigroups, which were introduced by Fountain [5] in the 1970s, are semigroups in which the cancellation properties of elements in general are encapsulated in the cancellation properties of idempotent elements. They form a common generalization of inverse semigroups and cancellative monoids, and their study is a key focus of the York School of semigroup theory.…”

Factorisable right adequate semigroups

“…Following (6) we say that a semigroup with or without an identity in which each i? *-class contains an idempotent and the idempotents commute is right adequate.…”