### Idempotent-Separating Extensions Of Inverse Semigroups

H. D'Alarcao
1969 Journal of the Australian Mathematical Society
Extensions of semigroups have been studied from two points of view; ideal extensions and Schreier extension. In this paper another type of extension is considered for the class of inverse semigroups. The main result (Theorem 2) is stated in the form of the classical treatment of Schreier extensions (see e.g. [7] ). The motivation for the definition of idempotentseparating extension comes primarily from G. B. Preston's concept of a normal set of subsets of a semigroup [6] . The characterization
more » ... f such extensions is applied to give another description of bisimple inverse co-semigroups, which were first described by N. R. Reilly [8] . The main tool used in the proof of Theorem 2 is Preston's characterization of congruences on an inverse semigroup [5] . For the standard terminology used, the reader is referred to [1]. Definitions and preliminaries Let A and B be inverse semigroups. A pair (S, f) where S is an inverse semigroup containing A as a subsemigroup and / is a homomorphism of S onto B such that /^(^( B ) ) = A (where for a semigroup T, E(T) = {e e T : e 2 = e}) is said to be an extension of A by B. If moreover / has the property that f(e) = f(g) implies e = g for e, ge E(A) then (5,/) is said to be an idempotent-separating extension of A by B. Let A and B be inverse semigroups and let (5, /) be an idempotent separating extension of A by B; by a transversal of B in 5 we mean a mapping g from B into S such that (i) f(g{b)) = b for all b e B. (ii) g(e) is the unique idempotent in f~x(e) for all e e E(B). (Sometimes {g(b) : b e B} will be itself called a transversal of B in S.)