Counting formulas for glued lattices
type: Item
instance of: scholarly article
title: Counting formulas for glued lattices
zbMATH Open document ID: 0552.06003
DOI: 10.1007/BF00383603
author: Klaus Reuter
published in: Order
publication date: 1985-01-01T00:00:00Z
review text: A binary relation \(\theta\) on a complete lattice L is called a tolerance relation, if it is reflexive and symmetric and \(x_ 1\theta y_ 1\), \(x_ 2\theta y_ 2\) imply \(x_ 1\vee x_ 2\theta y_ 1\vee y_ 2,\) \(x_ 1\wedge x_ 2\theta y_ 1\wedge y_ 2\) for any elements \(x_ 1\), \(x_ 2\), \(y_ 1\), \(y_ 2\) of L. For any \(a\in L\) the notation \(a_{\theta}=\wedge \{x\in L| a\theta x\}\), \(a^{\theta}=\vee \{x\in L| a\theta x\}\) is introduced. The interval \([a]_{\theta}=[a_{\theta},(a_{\theta})^{\theta}]\) is called a block of \(\theta\). The factor-lattice L/\(\theta\) is considered; its elements are blocks of \(\theta\). If any two blocks of \(\theta\) such that one covers another in L/\(\theta\) are disjoint, then \(\theta\) is called a glued tolerance relation. A formula enumerating the number of elements of L with exactly k lower (or upper) covers is proved. Some similar formulae are proved and a new proof of Dilworth's covering theorem is given.
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06B10
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 06B23
zbMATH DE Number: 3878990
zbMATH Keywords: complete lattice
zbMATH Keywords: tolerance relation
zbMATH Keywords: blocks
zbMATH Keywords: glued tolerance relation
zbMATH Keywords: covers
zbMATH Keywords: Dilworth's covering theorem
reviewed by: Bohdan Zelinka
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3878990