Tue, 29 March 2022, 1 pm MDT

The study of Maltsev conditions is a significant part of universal algebra, with classical characterizations of families of varieties (congruence permutable, distributive, modular...) and recent advanced results by Hobby, McKenzie, Kearnes, Kiss, among others. In particular, the interplay between distinct Maltsev conditions for congruence modular varieties has led to a refined theory for such varieties.

Recall that a Maltsev condition is, roughly, a statement of the form *"there are some $n$ and terms $t_1,...,t_n$ such that a certain finite set of equations hold"*. As we mentioned, many deep and sophisticated results are known about Maltsev conditions. On the other hand, when two conditions are compared, really little is known about the exact value of the smallest $n$ as above. For example, a simple observation by A. Day asserts that if some variety $V$ has $k$ Jónsson terms witnessing congruence distributivity, then $V$ has $2k-1$ Day terms witnessing congruence modularity. About fifty years ago Day asked whether this result is best possible, but, to the best of our knowledge, an exact solution is not yet known.

A deeper problem (asked by Lakser, Taylor, Tschantz in 1985) concerns the relative lengths of sequences of Day and Gumm terms characterizing congruence modularity. More recently, Kazda, Kozik, McKenzie, Moore provided still another characterization of congruence distributive and modular varieties by means of *"directed"* terms. Again, the exact relationships between the lengths of the sequences of terms is not known. A solution of the above problems is supposed to provide either interesting exotic examples of congruence modular and distributive varieties, or more refined structure theorems.

We shall present recent results about the above Day, LTT and KKMM problems, with an unexpected application to congruence distributive varieties.