On Independently Partitionable Sets of Semigroup Identities

A set $\Sigma$ of first-order sentences is said to be independently partitionable if there exists a partition $\Sigma=\bigcup_{n\in\mathbb N}\Sigma_n$ such that $\mathrm{var}\Sigma\ne\mathrm{var}\Sigma\setminus\Sigma_n$ for every $n\in\mathbb N$. A set $\Sigma$ of first-order sentences is said to be finitely independently partitionable if there exists a partition $\Sigma=\bigcup_{n\in\mathbb N}\Sigma_n$ such that $\Sigma_n$ is finite and $\mathrm{var}\Sigma\ne\mathrm{var}\Sigma\setminus\Sigma_n$ for every $n\in\mathbb N$.
We construct some varieties $\mathfrak X$, $\mathfrak Y$, and $\mathfrak Z$ of semigroups such that $\mathfrak X$ has no independently partitionable basis for identities, $\mathfrak Y$ has an independently partitionable basis but has no finitely independently partitionable basis for identities, and $\mathfrak Z$ has a finitely independently partitionable basis but has no independent basis for identities. We also present varieties $\mathfrak X$ and $\mathfrak Y$ of semigroups such that $\mathfrak X\subset\mathfrak Y$, $\mathfrak X$ and $\mathfrak Y$ possess independent bases for their identities, and $\mathfrak X$ has an independently partitionable basis but has no finitely independently partitionable basis for its identities in $\mathfrak Y$; moreover, none of subvarieties of $\mathfrak Y$ covers $\mathfrak X$.