Congruence Class Sizes in Finite Sectionally Complemented Lattices

Grätzer, G. ; Schmidt, E. T.

Canadian mathematical bulletin, Tome 47 (2004) no. 2, pp. 191-205

Résumé

The congruences of a finite sectionally complemented lattice $L$ are not necessarily uniform (any two congruence classes of a congruence are of the same size). To measure how far a congruence $\Theta $ of $L$ is from being uniform, we introduce Spec $\Theta $ , the spectrum of $\Theta $ , the family of cardinalities of the congruence classes of $\Theta $ . A typical result of this paper characterizes the spectrum $S=({{m}_{j}}|j<n)$ of a nontrivial congruence $\Theta $ with the following two properties: $$({{S}_{1}})\,\,\,\,2\le n\,\,\text{and }n\ne 3.\,\,\,\,$$ $$({{S}_{2}})\,\,\,2\le {{m}_{j}}\,\,\text{and}\,\,{{m}_{j}}\ne 3,\,\,\,\text{for}\,\text{all}\,j<n.$$

Détail

DOI : 10.4153/CMB-2004-019-3

Mots-clés : 06B10, 06B15, congruence lattice, congruence-preserving extension