On the Monoid of All Order Preserving Full Contractions with a Fixed Set

Abstract

In this paper, we consider the monoid Fix O C T ( [ n ] , D ) of all order-preserving full contraction mappings that fix a subset say D of a finite n -element chain { 1 , 2 , ⋯ , n } . We characterize regularity, Green’s relations, and starred Green’s relations, and show that this monoid is left adequate. Furthermore, we determine the cardinality of Fix O C T ( [ n ] , D ) , the ranks of its two-sided ideals, and demonstrate that the ranks of the two-sided ideals and their corresponding Rees quotients are equal. Moreover, we deduce the rank of the monoid Fix O C T ( [ n ] , D ) .