(m, n)-commutative semigroups

Abstract

In the last two chapters we deal with the (m, n)-commutative and the n (2) permutable semigroups, respectively. A semigroup is called an (m, n)-commutative semigroup if it satisfies the identity (x 1 ... x m )(y 1 ... y n ) (m and n are positive i n tegers). For a fixed integer n ≥ 2, a semigroup S is cailed an n (2)-permu table semigroup if, for any n-tuple (x 1, x 2, ..., x n ) of elements of S, there is a positive in teger t with 1 ≤ t ≤ n-1 such that x 1 x 2 ... x t x t + 1 ... x n = x t + 1... x n x 1...x t . First we deat with the (m, n)-co mmutative semigroups, because some results about them are necessary in the ex a m ina tio ns of n(2)- per mutable ones. In this chapter the (m, n)-commutative semigroups are examined. In the first part of the chapter we determine all couples (m, n) of positive integers m and n for which a semigroup is (m, n)-commutative. Since an (m, n)commutative semigroup S is (m′, n′)-commutative for every m′ ≥ m and n′ ≥ n, it is sufficient to know the function fs(n) = min{m : S is (m, n) — commutative}. As every (m, n)-commutative semigroup is (1, m + n)-commutative, fs is defined for all positive integers. We define a special function, the permutation function, and show that the functions Is are exactly the permutation functions. In the second part of the chapter, we show that every (m, n)-commutative semigroup is an E-k semigroup for some integer k ≥ Z. We also show that every (1, 2)-commutative semigroup is exponential. In the third part of the chapter, we deal with the semilattice decomposition of (m, n)-commutative semigroups.