The Jónsson distributivity spectrum (Q1652855)

Let \(\mathcal V\) be a variety. Define \(J_{\mathcal V}(n)\) to be the least \(m\in\mathbb N\) such that for any algebra \({\mathbf A}\in {\mathcal V}\) and any congruences \(\alpha,\beta\) of \(\mathbf A\) we have \[ \alpha\wedge\underbrace{(\beta\circ\gamma\circ\dots)}_{n\,\text{compositions}}\subseteq \underbrace{(\alpha\wedge\beta)\circ(\alpha\wedge\gamma)\circ\dots}_{m\,\text{compositions}} \] The paper studies the behavior of the function \(J_{\mathcal V}\) in congruence distributive varieties. Some notable results that the paper obtains: (1) For each \(n,m\), there is a strong Maltsev condition equivalent to \(J_{\mathcal V}(n)\leq m\), (2) for any \(m,\ell\in\mathbb N\), we have \(J_{\mathcal V}(m\ell)\leq J_{\mathcal V}(m)\ell\), (3) if \(\mathcal V\) is 3-distributive, then \(J_{\mathcal V}(n)\leq n\) for all \(n\geq 3\), and (4) if \(\mathcal V\) has \(m+1\) Day terms, then \(J_{\mathcal V}(2)\leq J_{\mathcal V}(1)+2m^2-2m-1\).