Injective linear transformations with equal gap and defect

Abstract

Suppose V is an infinite-dimensional vector space over a field F and let I(V) denote the inverse semigroup of all injective partial linear transformations on V. Given ß in I(V), we denote the domain and the range of ß by dom ß and im ß, respectively, and we call the cardinals g(ß)=codim(domß) and d(ß)=codim(imß) the `gap' and the `defect' of ß, respectively. In this paper, we study the semigroup A(V) of all injective partial linear transformations with equal gap and defect, and characterise Green's relations and ideals in A(V). This is analogous to work by Sanwong and Sullivan in 2009 on a similarly-defined semigroup for the set case, but we show that these semigroups are never isomorphic.