A problem on semigroups satisfying permutation identities

Let sigma be a nontrivial permutation of order n. A semigroup S is said to be sigma-permutable if x(1)x(2)...x(n)=x(sigma(1))x(sigma(2))...x(sigma(n)), for every (x(1),x(2),...,x(n)) is an element of S-n. A semigroup S is called (r, t)-commutative, where r, t are in N*, if x(1)...x(r)x(r+1)...x(r+t)=xr(+1)...x(r+t)x(1)...x(r), for every (x(1),x(2),...,x(r+t)) is an element of S-r+t. According to a result of M. Putcha and A. Yaqub ([11]), if sigma is a fixed-point-free permutation and S is a sigma-permutable semigroup then there exists k is an element of N* such that S is (1, k)-commutative. In [8], S. Lajos raises up the problem to determine the least k=k(n) is an element of N* such that, for every fixed-point-free permutation sigma of order n, every sigma-permutable semigroup is also (1, k)-commutative. In this paper this problem is solved for any n less than or equal to eight and also when n is any odd integer. For doing this we establish that if a semigroup satisfies a permutation identity of order n then inevitably it also satisfies some permutation identities of order n+1.