Regularity and Green's Relations for Generalized Semigroups of Transformations with Invariant Set

Lei Sun

Journal of Mathematics Research, 2018, vol. 10, issue 2, 24-28

Abstract: Let ${\cal T}_X$ be the full transformation semigroup on a set $X$. For $Y\subseteq X$, the semigroup $S(X,Y) =\{ f\in {\cal T}_X: f(Y)\subseteq Y\}$ is a subsemigroup of ${\cal T}_ X $. Fix an element $\theta\in S(X,Y)$ and for $f,g\in S(X,Y)$, define a new operation $*$ on $S(X,Y)$ by $f* g=f\theta g$ where $f\theta g$ denotes the produce of $g,\theta$ and $f$ in the original sense. Under this operation, the semigroup $S(X,Y)$ forms a semigroup which is called generalized semigroup of $S(X,Y)$ with the sandwich function $\theta$ and denoted by $S(X,Y,*_\theta)$. In this paper we first characterize the regular elements and then describe Green's relations for the semigroup $S(X,Y,*_\theta)$.

Keywords: generalized transformation semigroups; regular elements; Green¡¯s relations (search for similar items in EconPapers)
Date: 2018
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