On certain semigroups of full contraction maps of a finite chain

Authors : Goje Uba Garba, Muhammad Jamilu Ibrahim, Abdussamad Tanko Imam

Pages : 500-507

View : 12 | Download : 8

Publication Date : 9999-12-31

Article Type : Makaleler

Abstract :Let $X_{n}=\{1,2,\ldots,n\}$ with its natural order and let ${\cal T}_{n}$ be the full transformation semigroup on $X_{n}$. A map $\alpha\in{\cal T}_{n}$ is said to be order-preserving if, for all $x,y\in X_{n}$, $x\leq y\Rightarrow x\alpha\leq y\alpha$. The map $\alpha\in{\cal T}_{n}$ is said to be a contraction if, for all $x,y\in X_{n}$, $|x\alpha-y\alpha|\leq |x-y|$. Let ${\cal CT}_{n}$ and ${\cal OCT}_{n}$ denote, respectively, subsemigroups of all contraction maps and all order-preserving contraction maps in ${\cal T}_{n}$. In this paper we present characterisations of Green's relations on ${\cal CT}_{n}$ and starred Green's relations on both ${\cal CT}_{n}$ and ${\cal OCT}_{n}$.
Keywords : Full transformation, order-preserving, contraction, Green's relations, starred Green's relations