On Presentations of Semigroup of Transformations Restricted by an Equivalence
M. J. Ibrahim
Sule Lamido University, Kan-Hausa, Jigawa State, Nigeria.
D. A. Oluyori *
Ahmadu Bello University, Zaria, Kaduna State, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
For a non-empty set X denote the full transformation semigroup of X by T(X). Let \(\sigma\) be an equivalence relation on X and E(X, \(\sigma\)) denotes the semigroup (under composition) of all \(\alpha\) : X \(\mapsto\) X, such that \(\sigma\) \(\subseteq\) ker(\(\alpha\) ). Semigroup of transformations with restricted equivalence occur when we take all transformations whose kernel is contained in some fixed equivalence, E(X, \(\sigma\)). First, we found that E(X, \(\sigma\)) is a disjoint union copies of two generating sets. Next, we discuss the presentations, acts, subacts, direct products and bilateral semidirect product of the semigroup of transformation with restricted equivalence E(X, \(\sigma\)) and its application.
Keywords: Semigroups of transformations, generating set, presentations, direct product, bilateral semi-direct product