One-sided congruences on inverse semigroups

Abstract

By the kernel of a one-sided (left or right) congruence p on an inverse semigroup S, we mean the set of p-classes which contain idempotents of S. We provide a set of independent axioms characterizing the kernel of a one-sided congruence on an inverse semigroup and show how to reconstruct the one-sided congruence from its kernel. Next we show how to characterize those partitions of the idempotents of an inverse semigroup S which are induced by a one-sided congruence on S and provide a characterization of the maximum and minimum one-sided congruences on S inducing a given such partition. The final two sections are devoted to a study of indempotent-separating one-sided congruences and a characterization of all inverse semigroups with only trivial full inverse subsemigroups. A Green-Lagrange-type theorem for finite inverse semigroups is discussed in the fourth section. 1. Basic notions, terminology. We adhere throughout to the notation and terminology of A. H. Clifford and G. B. Preston [1]. Throughout the paper, S will always denote an inverse semigroup (i.e., for each a ES, there exists a unique element a~l E S suchthat a = aa~la and a-1 = a~1aa~1) and Es will denote the set of idempotents of S. The elementary properties of inverse semigroups may be found in [1]. In particular, we shall liberally use, without comment, the fact that Es is a semilattice (a commutative semigroup of idempotents) and that a~lEsa CES Va E S. We shall also use the fact that if S is an inverse semigroup then the Green's relations L and R on S are given by L = {(a, b) G S x S: a~xa = b~lb}