Permutation functions and (n, m)-commutativity of semigroups

Abstract

By [4], a semigroupS is called an (n, m)-commutative semigroup (n, m ∈ ℕ⁺, the set of all positive integers) if

$$x_1 x_2 \cdot \cdot \cdot x_n y_1 y_2 \cdot \cdot \cdot y_m = y_1 y_2 \cdot \cdot \cdot y_m x_1 x_2 \cdot \cdot \cdot x_n $$

holds for allx ₁,...,x _n ,y ₁,...,y _m ∈S It is evident that ifS is an (n, m)-commutative semigroup then it is (n′,m′)-commutative for alln′≥n andm′≥m. In this paper, for an arbitrary semigroupS, we determine all pairs (n, m) of positive integersn andm for which the semigroupS is (n, m)-commutative. In our investigation a special type of function mapping ℕ⁺ into itself plays an important role. These functions which are defined and discussed here will be called permutation functions.