The structure of elements in finite full transformation semigroups

The index and period of an element a of a finite semigroup are the smallest values of m ≥ 1 and r ≥ 1 such that am+r = am. An element with index m and period 1 is called an m-potent element. For an element α of a finite full transformation semigroup with index m and period r, a unique factorisation α = σβ such that Shift(σ) ∩ Shift(β) = ∅ is obtained, where σ is a permutation of order r and β is an m-potent. Some applications of this factorisation are given.