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 OE(X) = {f ∈ TE(X) : ∀x, y ∈ X, x ≤ y⟹f(x) ≤ f(y)} , where TE(X) is an E− preserving transformation semigroup. Obviously, OE(X) is a subsemigroup of TE(X), which is called an order‐preserving and equivalence‐preserving transformation semigroup. We fix an element θ in OE(X) and define a new operation ∘ on OE(X) by f∘g = fθg. Under the operation ∘, OE(X) forms a semigroup, which is called a generalized semigroup of OE(X) and is denoted by OE(X; θ). In this paper, we characterize the natural partial order on OE(X; θ), and the condition under which the two elements of OE(X; θ) are related to such natural partial order is also described. Furthermore, we investigate the elements of OE(X; 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.