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 PGY (X) = {α ∈ T(X) : α|Y ∈ G(Y )} where G(Y ) is the permutation group on Y . It is known that PGY (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 PGY (X) . Moreover, we prove some isomorphism theorems for PGY (X) . For finite sets, we investigate the cardinalities of PGY (X) and of its subsets of idempotents, and we also calculate their ranks.

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