Generalized commutativity of lattice-ordered groups II

Abstract

Commutative ${\ell}$-groups G (in which for all ${x, y \in G, xy = yx}$) were studied long ago. This was then generalized to the study of ${\ell}$-groups G in which for a given integer n and for all ${x, y \in G, x^{n}y^{n} = y^{n}x^{n}}$. It was then discovered that if for all ${x, y \in G}$, both ${x^{n}y^{n} = y^{n}x^{n}}$ and ${x^{m}y^{m} = y^{m}x^{m}}$ for two different integers m, n, then also ${x^{d}y^{d} = y^{d}x^{d}}$, where d is the greatest common divisor of m, n.

We will now generalize this to consider an ${\ell}$-group G in which for two fixed integers ${m, n, x^{m}y^{n} = y^{n}x^{m}}$ for all ${x, y \in G}$. Then we will generalize this to a set of more than two integers.

Finally, we will consider an even more general situation where one or both of the exponents are not fixed.