On Signed Full Transformation Semigroup of a Finite Set

Usman Mohammed Yusuf; Moses Anayo Mbah; Abimiku Alaku

doi:10.62050/fjst2025.v9n1.510

Authors

Usman Mohammed Yusuf Department of Mathematics, Federal University of Lafia Author
Dr. Mbah Moses A. Department of Mathematics, Federal University of Lafia, Lafia, Nasarawa State Author
Alaku A. Abimiku Department of Basic Science, College of Agriculture, Science and Technology, Lafia, Nasarawa State Author

DOI:

https://doi.org/10.62050/fjst2025.v9n1.510

Keywords:

Semigroup, Signed transformation, Order preserving transformation, Order decreasing transformation, Idempotents

Abstract

If we define [n] = {1,2,3,...,n} and [n^*] = {±1,±2,±3...,±n}. A map α: [n] → [n^*] is called a signed transformation on [n]. The collection of all these maps together with composition forms a semigroup called a signed transformation semigroup. Given that dom(α) = [n], the signed transformation semigroup will be called a signed full transformation semigroup on [n]. In this paper, we obtain formulas that count the number of elements in the semigroups of order decreasing, order preserving and order decreasing signed transformations on [n]. We equally do same for the sub-semigroup of the signed transformation semigroup consisting only of idempotents.

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