Application of Lattice Theory on Order-Preserving Full Transformation Semigroup Via Fixed Points: Application of Lattice Theory on Order-Preserving Full Transformation Semigroup Via Fixed Points

Risqot Ibrahim Garba; NAIMAT  BAKARE GATTA; Suleiman  Akinwunmi A.; Tajudeen Aliu O.

Authors

Ibrahim G. R. Department of Mathematics and Statistics, Kwara State University, Malete, P.M.B 1530, Ilorin, Kwara State, Nigeria.
Bakare, G. N. UNIVERSITY OF ILORIN,ILORIN
Akinwunmi. S. A. Department of Mathematics and Statistics, Federal University of Kashere, Gombe, Nigeria,
Aliu, O. T.

Keywords:

Order-preserving full transformation, Lattice, Order-pLeast elements, Greatest elements, Fixed Point.

Abstract

Let be a finite set, T_nbe full transformation semigroup and OT_nbe subsemigroup of T_nof all order-preserving full transformation semigroup. Let transformation α ∈ OT_n∀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.

Application of Lattice Theory on Order-Preserving Full Transformation Semigroup Via Fixed Points