Some Transformation Semigroups Admitting Nearing Structure

Ng. Danpattanamongkon, P. Udomkavanich, Y. Kemprasit

Abstract

Denote by T(X) and P(X) the full transformation semigroup andthe partial transformation semigroup on a nonempty setX, respectively. The semigroups T(X) and P(X) are known to admit a right nearring structure for any X and they admit a left nearring structure only the case that $|X| = 1$. We generalize these results to the semigroups T(X, Y ) and P(X, Y ) under composition where $\emptyset \neg Y \subset X, T(X, Y ) = {\alpha \in T(X)| ran \alpha \subset Y } and P(X, Y ) = {\alpha \in P(X) | ran \alpha \subset Y }$. We obtain the analogous results that T(X, Y ) and P(X, Y ) admit a right nearring structure for any$\emptyset \neg Y \subset X$ and $|Y | = 1$ is necessary and sufficient for them to admit a left nearring structure.