Amenably ordered inverse semigroups

Abstract

A partially ordered inverse semigroup S is said to be left amenably ordered if for each a, b ~ S, a < b implies ala ~ b-lb. Any inverse semigroup is amenably ordered, on both sides, under the natural partial order < defined by a ~< b if and only if a = eb for some idempotent e in S. On the other hand amenably ordered inverse semigroups have been considered by Blyth and McFadden in the context of regular Dubreil-Jacotin semigroups and Bosbaeh has considered such semigroups as division ordered semigroups. In this paper, we consider the structure of left amenably ordered inverse semigroups in detail. Particular emphasis is placed on those in which the imposed partial order extends the natural partial order. If E is any semilattice then TE the semigroup of isomorphisms between principal ideals of E can be turned into a left amenably ordered inverse semigroup. In general, the imposed partial order is different from the natural partial order and, if E is a chain, TE is a ^-semilatticed semigroup. It is shown that every left amenable partial order on an inverse semigroup S, which extends the natural partial order, can be described by means of a cone, analogous to that used to describe the partial order in a lattice ordered group. The order structure of those left amenable partially ordered inverse semigroups is investigated and it is shown that if such a semigroup is v-semilatticed and the imposed partial order extends the natural partial order then it is a lattice ordered semigroup S in which the mapping a ~-+ a-aa is a lattice homomorphism onto the idempotents, which form a distributive lattice. Under various hypotheses, it is shown that S is also a distributive lattice. It is shown that amenable partial orders on an inverse semigroup S which extend the natural partial order are determined by cones contained in the centralizer of the idempotents of S. If such a partial order is up directed then S is a semilattice of groups and the structure of S is determined in detail. These results generalize and extend results of Bosbach on amenably ordered inverse semigroups. Throughout the paper we shall assume familiarity with the theory of inverse semigroups which is described in Clifford and Preston, Algebraic Theory of Semigroups, Vol. 1, 2, Math Surveys 7, Amer. Math. Soc., Providence, R.I., 1961, 1967 or Howie, An Introduction to Semigroup Theory, Academic Press, London, 1976.