On certain order-preserving transformation semigroups
Jia Zhang, Yanfeng Luo · Communications in Algebra · 2016
Let X be a finite chain. Denote by 𝒪(X) the monoid of all order-preserving full transformations on X. For any nonempty convex subchain Y of X, let 𝒪¯(X,Y)={α∈𝒪(X)|Yα⊆Y}. Then 𝒪¯(X,Y) is a submonoid of 𝒪(X).In this paper, we characterize Green’s relations and Green’s *-relations on 𝒪¯(X,Y). Then we prove that 𝒪¯(X,Y) is abundant but not regular. Furthermore, we describe the regular elements of 𝒪¯(X,Y) and determine when 𝒪¯(X,Y) is a regular semigroup. In addition, we compute the cardinalities of E(𝒪¯(X,Y)), Reg(𝒪¯(X,Y)) and 𝒪¯(X,Y), respectively.