Notes on externally \((m,n)\)-commutative semigroups (Q1357257)

Let \(S\) be a semigroup and \(m\), \(n\) be two positive integers. A semigroup \(S\) is called \((m,n)\)-commutative if the identity \(xy=yx\) holds for every \(x\in S^m\) and \(y\in S^n\). Also \(S\) is called externally \((m,n)\)-commutative if the identity \(xzy=yzx\) holds for every \(x\in S^m\), \(y\in S^n\) and \(z\in S\). These two notions were introduced by \textit{S. Lajos} in two earlier papers [PU.M.A., Pure Math. Appl., Ser. A 1, No. 1, 59-65 (1990; Zbl 0718.20030); ibid. 2, No. 1/2, 67-72 (1991; Zbl 0737.20034)]. They are interesting particular cases of some more general notions, the \(n\)-permutability and the \(\sigma\)-permutability, where \(\sigma\) is a permutation of order \(n\) [see \textit{M. Guţan}, C. R. Acad. Sci., Paris, Sér. I 319, No. 1, 5-10 (1994; Zbl 0812.20034)]. In this short note connections between the externally \((m,n)\)-commutativity, the \((m,n)\)-commutativity and the commutativity of a semigroup are established. It is proved that if \(S\) is an externally \((1,n)\)-commutative semigroup then \(S\) is also \((1,n+2)\)-commutative and the semigroup \(S^{2+[n/2]}\) is commutative. Finally it is shown that the externally \((m,n)\)-commutativity implies the \((m,n+2)\)-commutativity.