The ideal structure of semigroups of transformations with restricted range

Abstract

Let Y be a fixed non-empty subset of a set X and let T(X,Y) denote the semigroup of all total transformations from X into Y . In 1975, Symons described the automorphisms of T(X,Y). Three decades later, Nenthein, Youngkhong and Kemprasit determined its regular elements, and more re- cently Sanwong, Singha and Sullivan characterised all maximal and minimal congruences on T (X, Y ). In 2008, Sanwong and Sommanee determined the largest regular subsemigroup of T(X,Y) when |Y| ≠1 and Y≠X; and using this, they described the Green’s relations on T (X, Y ). Here, we use their work to describe the ideal structure of T (X, Y ). We also correct the proof of the corresponding result for a linear analogue of T (X, Y ).