Infimal generators and dualities between complete lattices
type: Item
instance of: scholarly article
title: Infimal generators and dualities between complete lattices
zbMATH Open document ID: 0638.06006
DOI: 10.1007/BF01774294
author: Ivan Singer
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1987-01-01T00:00:00Z
review text: The aim of this paper is to give some applications of infimal generators of complete lattices in the sense of Kutateladze and Rubinov to the study of dualities between two complete lattices E and F, both in the general and in some particular cases (i.e. mappings \(\Delta: E\to F\) satisfying \(\Delta(\inf x_ i)=\sup \Delta (x_ i)\) for all \(\{x_ i\}_{i\in I}\subseteq E\) and all index sets I, including the empty set \(I=\emptyset)\). Moreover, the author introduces and studies the quasi- complement of an element of a complete lattice with respect to a family of infimal generators. Further he gives some additional results for \(E=(2^ X,\supseteq)\), \(F=(2^ W,\supseteq)\) and \(E=(\bar R^ X,\leq)\), \(F=(\bar R^ W,\leq)\) (where X and W are arbitrary sets), with suitable families of infimal generators. The author obtains some lattice-theoretic properties of the relations between dualities \(\Delta: (2^ X,\supseteq)\to (2^ W,\supseteq)\), binary relations \(\rho\leq X\times W\), polarities \(\pi(\rho): (2^ X,\supseteq)\to (2^ W,\supseteq)\), coupling functionals \(\phi\) :X\(\times W\to \bar R\) and Fenchel-Moreau conjugations \(c(\phi): (\bar R^ X,\leq)\to (\bar R^ W,\leq)\).
Mathematics Subject Classification ID: 06B23
zbMATH DE Number: 4039968
zbMATH Keywords: infimal generators of complete lattices
zbMATH Keywords: dualities between two complete lattices
zbMATH Keywords: quasi-complement
zbMATH Keywords: binary relations
zbMATH Keywords: polarities
zbMATH Keywords: coupling functionals
zbMATH Keywords: Fenchel-Moreau conjugations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4039968