On Signed Full Transformation Semigroup of a Finite Set
Usman Mohammed Yusuf, Moses Anayo Mbah, Abimiku Alaku · FULafia Journal of Science and Technology · 2025
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.