Regularity and Green’s relations for the semigroups of partial and full contractions of a finite chain
Bashir Ali, Abdullahi Umar, M. M. Zubairu · Scientific African · 2023
Let [n]={1,2,…,n} be a finite chain and let Pn be the semigroup of partial transformations on [n]. Let CPn={α∈Pn:(for allx,y∈Domα)|xα−yα|≤|x−y|} be the semigroup of partial contractions on [n]. Then CPn is a subsemigroup of Pn. In this paper, in Sections 2 and 3 we give a necessary and sufficient condition for an element in CPn to be regular and characterize all the Green’s relations in the semigroup CPn, respectively. In Sections 4 and 5 we obtain analogous results for the subsemigroups of order preserving and/or order reversing contractions, and that of full contractions, respectively. Lastly in section 6, we characterize all the maximal subgroups of these semigroups.