Products of idempotents in finite full transformation semigroups

Abstract

SynopsisIf |X| = n and α is a singular mapping in J(X), define c(α) to be the number of cyclic orbits of α and f(α) to be the number of fixed points. Then α is expressible as a product of n + c(α)−f(α) idempotents of rank n − 1, and no smaller number of idempotents of rank n − 1 will suffice. The maximum possible value of n + c(α)–f(α) is [3/2(n − 1)], which is thus a best possible global lower bound for the number of idempotents required to generate a singular element of J(X).