Groups, semilattices and inverse semigroups. I, II

Abstract

ABSTRACT. An inverse semigroup S is called proper if the equations ea = e = e2 together imply a2 = a for each a, e E S. In this paper a construction is given for a large class of proper inverse semigroups in terms of groups and partially ordered sets; the semigroups in this class are called /"-semigroups. It is shown that every inverse semigroup divides a P-semigroup in the sense that it is the image, under an idempotent separating homomorphism, of a full subsemigroup of a P-semigroup. Explicit divisions of this type are given for u-bisimple semigroups, proper bisimple inverse semigroups, semilattices of groups and Brandt semigroups.The algebraic theory of inverse semigroups has been the subject of a considerable amount of investigation in recent years. Much of this research has been concerned with the problem of describing inverse semigroups in terms of their semilattice of idempotents and maximal subgroups. For example, Clifford [2] described the structure of all inverse semigroups with central idempotents (semilattices of groups) in these terms. Reilfy [17] A homomorphism 9 of an inverse semigroup S into an inverse semigroup T is called idempotent separating if 9 is one-to-one on the idempotents of S; that is, e2 = e,f2 = f, e9 = f9 imply e = /. A congruence is idempotent separating if it is the congruence of an idempotent separating homomorphism. Howie [5] showed that u = {(a,b) E S X S: aTxea = b~xeb for all e2 = e E S} is the maximum idempotent separating congruence on an inverse semigroup S. Further, Munn [13] has shown that T = S/p is fundamental; that is, the identity is the only congruence on T contained in <=¥. The importance of fundamental inverse semigroups stems from the fact (Munn [11], [13]) that, if T is a

Some covering theorems for locally inverse semigroups

“…of an order ideal and subsemigroup of a semigroup which, in a sense, generalizes the construction of a P-semigroup, which has proved to be useful in describing inverse semigroups (McAlister [7], [8]), by replacing the group which acts on a semilattice by a completely simple semigroup. The homomorphism which arises in Pastijn's theorem is special in that the inverse image of each idempotent is a completely simple semigroup.…”

Factorizable inverse monoids

“…where is inclusion, P is E-unitary, and is a covering, that is, surjective and idempotent-separating; moreover, P= = G: So we have McAlister's Covering Theorem [24] that every inverse semigroup has an E-unitary cover. On the converse side, any covering : P !…”

On a Class of Lattice Ordered Inverse Semigroups

“…The result is essentially due to Saitô. Our contribution is to simplify the algebraic structure by using the structure theorem for E-unitary inverse semigroups which was not available to him [7]. Proof.…”

“…Proof. The inverse semigroup S admits an E-unitary inverse cover P G E [7]. Let F be a free group which has G as a homomorphic image and induce an action of F on from that of G on .…”