The Gumm level equals the alvin level in congruence distributive varieties

Absorption and directed Jónsson terms

We prove that every congruence distributive variety has directed Jónsson terms, and every congruence modular variety has directed Gumm terms. The directed terms we construct witness every case of absorption witnessed by the original Jónsson or Gumm terms. This result is equivalent to a pair of claims about absorption for admissible preorders in CD and CM varieties, respectively. For finite algebras, these absorption theorems have already seen significant applications, but until now, it was not clear if the theorems hold for general algebras as well. Our method also yields a novel proof of a re…

Relation identities in 3-distributive varieties

Let $α$, $β$, $γ, \dots$ $Θ$, $Ψ, \dots$ $R$, $S$, $T, \dots$ be variables for, respectively, congruences, tolerances and reflexive admissible relations. Let juxtaposition denote intersection. We show that if the identity $α(β\circ Θ) \subseteq αβ\circ αΘ\circ αβ$ holds in a variety $\mathcal {V}$, then $\mathcal {V}$ has a majority term, equivalently, $\mathcal {V}$ satisfies $ α(β\circ γ) \subseteq αβ\circ αγ$. The result is unexpected, since in the displayed identity we have one more factor on the right and, moreover, if we let $Θ$ be a congruence, we get a condition equivalent to $3$-distr…

Day定理在$n$为偶数时是精确的

Day's Theorem is sharp for $n$ even

该研究解决了Maltsev条件相对长度问题，肯定回答了Day五十多年前的经典问题，证明当n为偶数时Day关于构造同余模项数量的结果精确，并处理多种项类型，结果适用于局部有限簇。