Prime Maltsev Conditions and Congruence n-Permutability

Abstract

For $n\geq2$, a variety $\mathcal{V}$ is said to be congruence $n$-permutable if every algebra $\mathbf{A}\in\mathcal{V}$ satisfies $\alpha\circ^n\beta=\beta\circ^n\alpha$, for all $\alpha,\beta\in \Con(\mathbf{A})$. Furthermore, given any algebra $\mathbf{A}$ and $k\geq1$, a $k$-dimensional Hagemann relation on $\mathbf{A}$ is a reflexive compatible relation $R\subseteq A\times A$ such that $R^{-1}\not\subseteq R\circ^k R$. A famous result of J. Hagemann and A. Mitschke shows that a variety $\mathcal{V}$ is congruence $n$-permutable if and only if $\mathcal{V}$ has no member carrying an $(n-1)$-dimensional Hagemann relation: by using this criterion, we provide another Maltsev characterization of congruence $n$-permutability, equivalent to the well-known Schmidt's and Hagemann-Mitschke's (\cite{HagMit}) term-based descriptions.

We further establish that the omission by varieties of certain special configurations of Hagemann relations induces the satisfaction of suitable Maltsev conditions. These omission properties may be used to characterize congruence $n$-permutable idempotent varieties for some $n\geq2$, congruence 2-permutable idempotent varieties and congruence 3-permutable locally finite idempotent varieties, yielding that the following are prime Maltsev conditions: \begin{enumerate} \item congruence $n$-permutability for some $n\geq2$ with respect to idempotent varieties; \item congruence 2-permutability with respect to idempotent varieties; \item congruence 3-permutability with respect to locally finite idempotent varieties. \end{enumerate}

Finally, we focus on the analysis of a family of strong Maltsev conditions, which we denote by $\{\mathcal{D}_n:2\leq n<\omega\}$, such that any variety $\mathcal{V}$ is congruence $n$-permutable whenever $\mathcal{D}_n$ is interpretable in $\mathcal{V}$. Among various other properties, we also show that the $\mathcal{D}_n$'s with odd $n\geq3$ generate decomposable strong Maltsev filters in the lattice of interpretability types.