Congruence Class Sizes in Finite Sectionally Complemented Lattices

http://dx.doi.org/10.4153/CMB-2004-019-3
Canad. Math. Bull. 47(2004), 191-205

  Published:2004-06-01
 Printed: Jun 2004

The congruences of a finite sectionally complemented lattice $L$ are not necessarily \emph{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 \emph{spectrum} of $\Theta$, the family of cardinalities of the congruence classes of $\Theta$. A typical result of this paper characterizes the spectrum $S = (m_j \mid j < n)$ of a nontrivial congruence $\Theta$ with the following two properties: \begin{enumerate}[$(S_2)$] \item[$(S_1)$] $2 \leq n$ and $n \neq 3$. \item[$(S_2)$] $2 \leq m_j$ and $m_j \neq 3$, for all $j

Keywords:

congruence lattice, congruence-preserving extension congruence lattice, congruence-preserving extension

MSC Classifications:

06B10, 06B15 show english descriptions Ideals, congruence relations Representation theory 06B10 - Ideals, congruence relations 06B15 - Representation theory