Natural Partial Order on Generalized Semigroups of Transformation Semigroups That Preserve Order and an Equivalence Relation
Meiqing Qin, Xuerong Fu · Journal of Mathematics · 2025
Let X be a finite total order set and E be a convex equivalence relation on X . We denote that O E ( X ) = { f ∈ T E ( X ) : ∀ x , y ∈ X , x ≤ y ⟹ f ( x ) ≤ f ( y )} , where T E ( X ) is an E − preserving transformation semigroup. Obviously, O E ( X ) is a subsemigroup of T E ( X ), which is called an order‐preserving and equivalence‐preserving transformation semigroup. We fix an element θ in O E ( X ) and define a new operation ∘ on O E ( X ) by f ∘ g = f θ g . Under the operation ∘, O E ( X ) forms a semigroup, which is called a generalized semigroup of O E ( X ) and is denoted by O E ( X ; θ ). In this paper, we characterize the natural partial order on O E ( X ; θ ), and the condition under which the two elements of O E ( X ; θ ) are related to such natural partial order is also described. Furthermore, we investigate the elements of O E ( 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.