On products of 3-paths in finite full transformation semigroups

Abstract

Let Singn denotes the semigroup of all singular self-maps of a finite set Xn={1,2, . . . , n}. A map α∈Singn is called a 3-path if there are i, j, k∈Xn such that iα=j,jα=k and xα=x for all x∈Xn\ {i, j}. In this paper, we described aprocedure to factorise each α∈Singn into a product of 3-paths. The length of each factorisation, that is the number of factors in eachfactorisation, is obtained to be equal to ⌈12(g(α)+m(α))⌉, where g(α) is known as the gravity of α and m(α) is a parameter introduced inthis work and referred to as the measure of α. Moreover, we showed that Singn⊆P[n−1], where P denotes the set of all 3-paths in Singn and P[k]=P∪P2∪ ··· ∪Pk.