An Application of the Natural Partial Order Relation on the Generalized Semigroup of Transformation With Invariant Set

Let X be a nonempty set and TX be the full transformation semigroup on X. For ∅≠Y⊆X, the semigroup of all transformations that leave Y invariant in TX is denoted as SX,Y=f∈TX:fY⊆Y. We fix an element θ∈SX,Y and specify an operation ∗ on SX,Y by f∗g=fθg, where fθg denotes the produce of θ,g and f in the general sense. Under the operation, ∗SX,Y formes a semigroup which is called generalized semigroup and denoted as SX,Y,∗θ. In this paper, we first endow the semigroup SX,Y,∗θ with a partial order relation and characterize the conditions under which two elements are related with respect to this partial order relation. Then, we describe the elements of the semigroup that are compatible with respect to this relation as well as extremely maximal (minimal) elements. Finally, we study the application of the partial order relation on the generalized semigroup SX,Y,∗θ in cryptography and computer science.