Regular elements of some transformation semigroups

Abstract. The full transformation semigroup on a nonempty set X is denoted by T (X). It is well-known that T (X) is a regular semigroup, that is, for every α ∈ T (X), α = αβα for some β ∈ T (X). The subsemigroups of T (X) we consider are T (X, Y ) and T (X, Y ) with ∅ 6= Y ⊆ X where T (X, Y ) = {α ∈ T (X) | ran α ⊆ Y } and T (X, Y ) = {α ∈ T (X) | Y α ⊆ Y }. Then T (X, Y ) ⊆ T (X, Y ). In 1966, K.D. Magill has studied the semigroup T (X, Y ) while J.S.V. Symons has studied the semigroup T (X, Y ) in 1975. In this paper, we characterize regular elements of both T (X, Y ) and T (X, Y ). These results are then applied to determine the numbers of regular elements in T (X, Y ) and T (X, Y ) for a nite set X. The numbers are given in terms of |X|, |Y | and their related Stirling numbers.