Natural Partial Order on Generalized Semigroups of Transformation Semigroups That Preserve Order and an Equivalence Relation

Let X be a finite total order set and E be a convex equivalence relation on X. We denote that OEX=f∈TEX:∀x,y∈X,x≤y⟹fx≤fy , where TEX is an E− preserving transformation semigroup. Obviously, OEX is a subsemigroup of TEX, which is called an order-preserving and equivalence-preserving transformation semigroup. We fix an element θ in OEX and define a new operation ∘ on OEX by f∘g=fθg. Under the operation ∘, OEX forms a semigroup, which is called a generalized semigroup of OEX and is denoted by OEX;θ. In this paper, we characterize the natural partial order on OEX;θ, and the condition under which the two elements of OEX;θ are related to such natural partial order is also described. Furthermore, we investigate the elements of OEX;θ that are compatible with this partial order and find out the maximal (minimal) elements. This study not only contributes to a deeper understanding of the internal structure of semigroups and the interactions between elements but can also be used to analyze the optimal path selection in graph theory and optimize traffic distribution problems in networks.