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Congruence Class Sizes in Finite Sectionally Complemented Lattices

  Published:2004-06-01
 Printed: Jun 2004
  • G. GrĂ€tzer
  • E. T. Schmidt
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Abstract

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
 

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