INJECTIVE TRANSFORMATIONS WITH EQUAL GAP AND DEFECT

Abstract

AbstractSuppose that X is an infinite set and I(X) is the symmetric inverse semigroup defined on X. If α∈I(X), we let dom α and ran α denote the domain and range of α, respectively, and we say that g(α)=|X/dom α| and d(α)=|X/ran α| is the ‘gap’ and the ‘defect’ of α, respectively. In this paper, we study algebraic properties of the semigroup $A(X)=\{\alpha \in I(X)\mid g(\alpha )=d(\alpha )\}$. For example, we describe Green’s relations and ideals in A(X), and determine all maximal subsemigroups of A(X) when X is uncountable.