Join-semilattices with two-dimensional congruence amalgamation

Friedrich Wehrung¹

¹ CNRS, UMR 6139 Département de Mathématiques, BP 5186 Université de Caen, Campus 2 14032 Caen Cedex, France

Colloquium Mathematicum, Tome 93 (2002) no. 2, pp. 209-235.

Résumé

We say that a $\langle\vee,0\rangle$-semilattice $S$ is conditionally co-Brouwerian if (1) for all nonempty subsets $X$ and $Y$ of $S$ such that $X\leq Y$ (i.e. $x\leq y$ for all $\langle{x,y}\rangle\in X\times Y$), there exists $z\in S$ such that $X\leq z\leq Y$, and (2) for every subset $Z$ of $S$ and all $a, b\in S$, if $a\leq b\vee z$ for all $z\in Z$, then there exists $c\in S$ such that $a\leq b\vee c$ and $c\leq Z$. By restricting this definition to subsets $X$, $Y$, and $Z$ of less than $\kappa$ elements, for an infinite cardinal $\kappa$, we obtain the definition of a conditionally $\kappa$-co-Brouwerian $\langle\vee,0\rangle$-semilattice.We prove that for every conditionally co-Brouwerian lattice $S$ and every partial lattice $P$, every $\langle\vee,0\rangle$-homomorphism $\varphi: \mathop{\rm Con}\nolimits_{\rm c} P\to S$ can be lifted to a lattice homomorphism $f: P\to L$ for some relatively complemented lattice $L$. Here, $\mathop{\rm Con}\nolimits_{\rm c} P$ denotes the $\langle\vee,0\rangle$-semilattice of compact congruences of $P$.We also prove a two-dimensional version of this result, and we establish partial converses of our results and various of their consequences in terms of congruence lattice representation problems. Among these consequences, for every infinite regular cardinal $\kappa$ and every conditionally $\kappa$-co-Brouwerian $S$ of size $\kappa$, there exists a relatively complemented lattice $L$ with zero such that $\mathop{\rm Con}_{\rm c}L\cong S$.

Détail

DOI : 10.4064/cm93-2-2

Keywords: say langle vee rangle semilattice conditionally co brouwerian nonempty subsets leq leq langle rangle times there exists leq leq every subset leq vee there exists leq vee leq restricting definition subsets kappa elements infinite cardinal kappa obtain definition conditionally kappa co brouwerian langle vee rangle semilattice prove every conditionally co brouwerian lattice every partial lattice every langle vee rangle homomorphism varphi mathop con nolimits lifted lattice homomorphism relatively complemented lattice here mathop con nolimits denotes langle vee rangle semilattice compact congruences prove two dimensional version result establish partial converses results various their consequences terms congruence lattice representation problems among these consequences every infinite regular cardinal kappa every conditionally kappa co brouwerian size kappa there exists relatively complemented lattice zero mathop con cong