On Signed Full Transformation Semigroup of a Finite Set

Résumé

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.