Congruence-Permutable, Congruence-Preserving Extensions of Lattices

Abstract

This chapter is intended to illustrate how to use CLL for solving an open problem of lattice theory, although the statement of that problem does not involve lifting diagrams with respect to functors. We set $\alpha\beta:=\{(x,z)\in X \times X \mid (Ey\in X)(x,y)\in \alpha {\rm and}(y,z)\in \beta\}$, for any binary relations a and ß on a set X, and we say that an algebra A is congruence-permutable if aß = ßa for any congruences a and ß of A. The problem in question is: Does every lattice of cardinalityN1 have a congruence-permutable, congruence-preserving extension? This problem is part of Problem 4 in the survey paper Tůma and Wehrung [59] but it was certainly known before. Due to earlier work by Ploščica et al. [49], the corresponding negative result for free lattices on ?2 gen- erators was already known. The countable case is still open, although it is proved in Grätzer et al. [30] that every countable, locally finite (or even “locally congruence-finite”) lattice has a relatively complemented (thus congruence-permutable) congruence-preserving extension. The finite case is solved in Tischendorf [57], where it is proved that every finite lattice embeds congruence-preservingly into some finite atomistic (thus congruence- permutable) lattice. This result is improved in Grätzer and Schmidt [32], where the authors prove that every finite lattice embeds congruence- preservingly into some sectionally complemented finite lattice. Most of Chap. 5 will consist of checking one after another the various assumptions that need to be satisfied in order to be able to use CLL. Most of these verifications are elementary. In that sense (i.e., considering the verification of the assumptions underlying CLL as tedious but elementary), the hard core of the solution to the problem above consists of the unliftable family of squares presented in Lemma 5.3.1.