Abstract
Let S be a semigroup. An element a of S is called a right [left] magnifying element if there exists a proper subset M of S satisfying S = M a [ S = a M ] . Let E be an equivalence relation on a nonempty set X. In this paper, we consider the semigroup P ( X , E ) consisting of all E-preserving partial transformations, which is a subsemigroup of the partial transformation semigroup P ( X ) . The main propose of this paper is to show the necessary and sufficient conditions for elements in P ( X , E ) to be right or left magnifying.
Highlights
An element a of a semigroup S is a right [left] magnifying element in S if there exists a proper subset M of S satisfying S = Ma [S = aM ]
Theorem 1. α is right magnifying in P( X, E) if and only if α is onto, for any ( x, y) ∈ E, there exists ( a, b) ∈ E
Α is right magnifying in a semigroup P( X ) if and only if α is onto and either dom α 6= X or dom α = X and α is not one-to-one
Summary
An element a of a semigroup S is a right [left] magnifying element in S if there exists a proper subset M of S satisfying S = Ma [S = aM ]. A year later, he showed in [6] that every semigroup which contains magnifying elements is factorizable; this solved a problem raised by Catino and Migliorini. In 2013, Huisheng and Weina [15] studied naturally orderd semigroups of partial transformations preserving an equivalence relation. Chinram and Baupradist have lately investigated right and left magnifying elements in some generalized transformation semigroups in [16,17]. The semigroup of partial transformations preserving the equivalence relation E. is exactly a subsemigroup of P( X ). We study right and left magnifying elements in P( X, E) and conclude necessary and sufficient conditions for elements of P( X, E) to be left or right magnifying
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