Some Transformation Semigroups Admitting Nearing Structure
Ng. Danpattanamongkon, Patanee Udomkavanich, Yupaporn Kemprasit · Thai Journal of Mathematics · 2012
Denote by T ( X ) and P ( X ) the full transformation semigroup and the partial transformation semigroup on a nonempty set X , 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 $\emptyset eg Y \subset X , T ( X, Y ) = {\alpha \in T ( X )| ran \alpha \subset Y } and P( X, Y ) = {\alpha \in P ( X ) | ran \alpha \subset Y }$ . We obtain the analogous results that T ( X, Y ) and P ( X, Y ) admit a right nearring structure for any $\emptyset eg Y \subset X$ and $| Y | = 1$ is necessary and sufficient for them to admit a left nearring structure.