On the ranks of semigroups of transformations on a finite set with restricted range

Abstract

Let PT (X) be the semigroup of all partial transformations on X, T (X) and I(X) be the subsemigroups of PT (X) of all full transformations on X and of all injective partial transformations on X, respectively. Given a non-empty subset Y of X, let PT (X; Y ) = {α ∈ PT (X) : Xα ⊆ Y}, T (X; Y ) = PT (X; Y ) \ T (X) and I(X; Y ) = PT (X; Y ) \ I(X). In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined Green's relations of T (X; Y ). In this paper, we present analogous results for both PT (X; Y ) and I(X; Y ). For a finite set X with jXj - 3, the ranks of PT (X) = PT (X;X), T (X) = T (X;X) and I(X) = I(X;X) are well known to be 4, 3 and 3, respectively. In this paper, we also compute the ranks of PT (X; Y ), T (X; Y ) and I(X; Y ) for any proper non-empty subset Y of X. © 2014 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.