THE IDEAL STRUCTURE OF SEMIGROUPS OF TRANSFORMATIONS WITH RESTRICTED RANGE

Abstract

Abstract Let Y be a fixed nonempty 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 recently Sanwong, Singha and Sullivan characterized 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 ) .