Infimal generators and dualities between complete lattices
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 \(c(\phi): (\bar R^ X,\leq)\to (\bar R^ W,\leq)\).
scientific article; zbMATH DE number 4039968
scientific article