Regulations and idempotent properties for finite ordered-preserving sandwich semigroup OT(X,Y;θ)

Let X and Y be arbitrary nonempty order sets,OT(X,Y) be the set of mappings from X to Y,θ be arbitrary but fixed mapping from Y to X for any α,β∈OT(X,Y),the operation in OT(X,Y) is defined by α°β=αθβ,where αθβ is the production of mappings.Then OT(X,Y) forms a semigroup called sandwich semigroup and denoted by OT(X,Y).The sandwich semigroup OT(X,Y) is called finite preserving order sandwich semigroup when both X and Y are finite sets and |X|1,|Y|1.In this paper,we discuss the regulations and idempotent properties of OT(X,Y;θ).