Congruence modularity at 0

Abstract

Given a variety V with a constant 0 in its type and a lattice identity p ≤ q, we say that p ≤ q holds for congruences in V at 0 if the p-block of 0 is included in the q-block of 0 for all substitutions of congruences of V-algebras for the variables of p and q. Varieties that are congruence modular at 0 are characterized by a Mal'tsev condition. This result generalizes the classical characterization of congruence modularity by Day terms.

Modular and semimodular lattices

“…Similarly to congruence modularity, we call A congruence modular at 0, if the above defined lattice is modular. Proving the conjecture of Ivan Chajda, we show that congruence modularity at 0 can be characterized by a Mal'cev condition, see [77].…”

“…However, if a lattice is of finite length and both itself and its dual are semimodular then it is also modular. The three chapters of my dissertation are based on the papers [27,78] and [77].…”

“…Then we gave a Mal'cev condition in [77] that characterizes congruence modularity at 0. Note that Jónsson's and Day's characterization of congruence distributivity and congruence modularity follows from the characterization of congruence distributivity and congruence modularity at 0, cf.…”

“…Chajda sejtését igazolva megmutatom, hogy a 0-nál vett kongruencia-modularitás jellemezhető Mal'cev feltétellel [77].…”

“…After defining the concept of a Mal'cev condition, we show that a classical result of Alan Day [28], which says that congruence modular varieties can be defined by a Mal'cev condition, can be generalized for lattices observed by Chajda. This part is based on [77].…”