Day's Theorem is sharp for nn even

Both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from J\'onsson terms $t_0, \dots, t_n$ witnessing congruence distributivity it is possible to construct terms $u_0, \dots, u _{2n-1}$ witnessing congruence modularity. We show that Day's result about the number of such terms is sharp when $n$ is even. We also deal with other kinds of terms, such as alvin, Gumm, directed, specular, mixed and defective. All the results hold also when restricted to locally finite varieties. We introduce some families of congruence distributive varieties and characterize many congruence identities they satisfy.Comment: v.2, some improvements and some corrections, particularly in Section 9 v.3, a few further improvements, corrections simplification

Congruence lattices of semilattices

The main result of this paper is that the class of congruence lattices of semilattices satisfies no nontrivial lattice identities. It is also shown that the class of subalgebra lattices of semilattices satisfies no nontrivial lattice identities. As a consequence it is shown that if V is a semigroup variety all of whose congruence lattices satisfy some fixed nontrivial lattice identity, then all the members of V are groups with exponent dividing a fixed finite number

On the J\'onsson distributivity spectrum

Suppose throughout that $\mathcal V$ is a congruence distributive variety. If $m \geq 1$, 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 $J _{ \mathcal V} (m \ell ) \leq k \ell$, 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 $J _{ \mathcal V} (m) \leq m$, for every $m \geq 3$ (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 $J _{ \mathcal V} (2) \leq J _{ \mathcal V} (1) + 2m^2-2m -1$. Various problems are stated at various places.Comment: v. 4, added somethin

The possible values of critical points between varieties of lattices

We denote by Conc(L) the semilattice of all finitely generated congruences of a lattice L. For varieties (i.e., equational classes) V and W of lattices such that V is contained neither in W nor its dual, and such that every simple member of W contains a prime interval, we prove that there exists a bounded lattice A in V with at most aleph 2 elements such that Conc(A) is not isomorphic to Conc(B) for any B in W. The bound aleph 2 is optimal. As a corollary of our results, there are continuum many congruence classes of locally finite varieties of (bounded) modular lattices

Lifting retracted diagrams with respect to projectable functors

We prove a general categorical theorem that enables us to state that under certain conditions, the range of a functor is large. As an application, we prove various results of which the following is a prototype: If every diagram, indexed by a lattice, of finite Boolean (v,0)-semilattices with (v,0)-embeddings, can be lifted with respect to the \Conc functor on lattices, then so can every diagram, indexed by a lattice, of finite distributive (v,0)-semilattices with (v,0-embeddings. If the premise of this statement held, this would solve in turn the (still open) problem whether every distributive algebraic lattice is isomorphic to the congruence lattice of a lattice. We also outline potential applications of the method to other functors, such as the $R\mapsto V(R)$ functor on von Neumann regular rings