Counting formulas for glued lattices
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.
scientific article; zbMATH DE number 3878990
scientific article