On the Jónsson distributivity spectrum

Abstract

Suppose throughout that $\mathcal{V}$ is a congruence distributive variety. If $m\ge 1$, let $J_{\mathcal{V}}(m)$ be the smallest natural number $k$ such that the congruence identity $\alpha (\beta \circ \gamma \circ \beta \dots )\subseteq \alpha \beta \circ \alpha \gamma \circ \alpha \beta \circ \dots$ holds in $\mathcal{V}$, with $m$ occurrences of $\circ$ on the left and $k$ occurrences of $\circ$ on the right. We show that if $J_{\mathcal{V}}(m)=k$, then $J_{\mathcal{V}}(m\ell )\le k\ell$, for every natural number $\ell$. The key to the proof is an identity which, through a variety, is equivalent to the above congruence identity, but involves also reflexive and admissible relations. If $J_{\mathcal{V}}(1)=2$, that is, $\mathcal{V}$ is $3$-distributive, then $J_{\mathcal{V}}(m)\le m$, for every $m\ge 3$ (actually, a more general result is presented which holds even in nondistributive varieties). If $\mathcal{V}$ is $m$-modular, that is, congruence modularity of $\mathcal{V}$ is witnessed by $m+1$ Day terms, then $J_{\mathcal{V}}(2)\le J_{\mathcal{V}}(1)+2m^2-2m-1$. Various problems are stated at various places.

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References in corpus (2)

• Relation identities equivalent to congruence modularity
• Unions of admissible relations

Cited by in corpus (6)

• Relation identities in 3-distributive varieties
• On the number of terms witnessing congruence modularity
• Relation identities equivalent to congruence modularity
• The Gumm level equals the alvin level in congruence distributive varieties
• The distributivity spectrum of Baker's variety
• A short way to directed Jónsson terms