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2.44 (1966)

Solved

Let $\mathfrak{A}$ and $\mathfrak{B}$ be subvarieties of a variety of groups $\mathfrak{M}$; then $(\mathfrak{A}\mathfrak{B}) \cap \mathfrak{M}$ is called the $\mathfrak{M}$-product of $\mathfrak{A}$ by $\mathfrak{B}$, where $\mathfrak{A}\mathfrak{B}$ is the usual product. Does there exist a non-abelian variety $\mathfrak{M}$ with an infinite lattice of subvarieties and commutative $\mathfrak{M}$-multiplication?

Contributor: A. I. Mal’cev

Progress

Yes, there does; for example, $\mathfrak{M} = \mathfrak{A}_p\mathfrak{A}_p$, where $\mathfrak{A}_p$ is the variety of all abelian groups of prime exponent $p$ (Yu. M. Gorchakov, Talk of June 21, 1967, Krasnoyarsk).

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