Application of Lattice Theory on Order-Preserving Full Transformation

Abstract

Let be a finite set, Tn be full transformation semigroup and OTn be subsemigroup of Tn of all order-preserving full transformation semigroup. Let transformation α ∈ OTn ∀x, y ∈ α, if x ≤ y then α(x) ≤ α(y), then α is called order-preserving transformation. This paper focuses on the notion of fixed points which are elements that remain unchanged under this transformation. The existence of fixed points were explored and emphasizing their role in establishing a lattice structure. The lattice of fixed points exhibits two essential operations: meet and join. These operations enable us to compute the greatest and least elements of fixed points. Beyond pure mathematics, the study of fixed points and their lattice structure has applications in dynamical systems, economics, computer science, and several other domains, making it both a theoretical and practical subject.