Injective transformations with equal gap and defect

Abstract

Suppose that X is an infinite set and I(X) is the symmetric inverse semigroup defined on X. If α ε I(X), we let dom α and ran α denote the domain and range of α , respectively, and we say that g(α)=|X/domα| and d(α)=|X/ranα| is the gap and the defect of , respectively. In this paper, we study algebraic properties of the semigroup $A(X)=\{α I(X) g(α )=d(α). For example, we describe Greens relations and ideals in A(X), and determine all maximal subsemigroups of A(X) when X is uncountable. Copyright © Australian Mathematical Society 2009.