A problem on semigroups satisfying permutation identities

Abstract

Let σ be a nontrivial permutation of ordern. A semigroupS is said to be σ-permutable ifx ₁ x ₂ ...x _n =x _σ(1) x _σ(2) ...x _σ(n) , for every (x ₁ ,x ₂,...,x _n )∈S ⁿ. A semigroupS is called(r,t)-commutative, wherer,t are in ℕ^*, ifx ₁ ...x _r x _r+1 ...x _r+t =x _r+1 ...x _r+t x ₁ ...x _r , for every (x ₁ ,x ₂,...,x _r+t ∈S ^r+t. According to a result of M. Putcha and A. Yaqub ([11]), if σ is a fixed-point-free permutation andS is a σ-permutable semigroup then there existsk ∈ ℕ^* such thatS is (1,k)-commutative. In [8], S. Lajos raises up the problem to determine the leastk=k(n) ∈ ℕ^* such that, for every fixed-point-free permutation σ of ordern, every σ-permutable semigroup is also (1,k)-commutative. In this paper this problem is solved for anyn less than or equal to eight and also whenn is any odd integer. For doing this we establish that if a semigroup satisfies a permutation identity of ordern then inevitably it also satisfies some permutation identities of ordern+1.