INJECTIVE TRANSFORMATIONS WITH EQUAL GAP AND DEFECT
Jintana Sanwong, Robert Patrick Sullivan · Bulletin of the Australian Mathematical Society · 2009
Abstract Suppose thatXis an infinite set andI(X) is the symmetric inverse semigroup defined onX. Ifα∈I(X), we letdom αand ran αdenote the domain and range ofα, respectively, and we say thatg(α)=|X/dom α| andd(α)=|X/ran α| is the ‘gap’ and the ‘defect’ ofα, 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 inA(X), and determine all maximal subsemigroups ofA(X) whenXis uncountable.