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


  •  Lei Sun    

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 $S(X,Y,*_\theta)$.


This work is licensed under a Creative Commons Attribution 4.0 License.
  • Issn(Print): 1916-9795
  • Issn(Onlne): 1916-9809
  • Started: 2009
  • Frequency: bimonthly

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