Some Properties of the semigroup PGY (X): Green’s relations, ideals, isomorphism theorems and ranks.

Let T(X) be the full transformation semigroup on the set X . For a fixed nonempty subset Y of X, let PG_Y (X) = {α ∈ T(X) : α|Y ∈ G(Y )} where G(Y ) is the permutation group on Y . It is known that P GY (X) is a regular subsemigroup of T(X). In this paper, we give a simpler description of Green’s relations and characterize the ideals of P GY (X). Moreover, we prove some isomorphism theorems for PGY (X). For finite sets, we investigate the cardinalities of PG_Y (X) and of its subsets of idempotents, and we also calculate their ranks. [ABSTRACT FROM AUTHOR]