Relational representations of algebraic lattices and their applications

Abstract

In this paper, we define the concepts of strongly regular relation, finitely strongly regular relation, and generalized finitely strongly regular relation, and get the relational representations of strongly algebraic, hyperalgebraic, and quasi-hyperalgebraic lattices. The main results are as follows: (1) a binary relation ρ:X⇀Y\rho :X\hspace{0.33em}\rightharpoonup \hspace{0.33em}Y is strongly regular if and only if the complete lattice (Φρ(X),⊆)\left({\Phi }_{\rho }\left(X),\subseteq ) is a strongly algebraic lattice; (2) a binary relation ρ:X⇀Y\rho :X\hspace{0.33em}\rightharpoonup \hspace{0.33em}Y is finitely strongly regular if and only if (Φρ(X),⊆)\left({\Phi }_{\rho }\left(X),\subseteq ) is a hyperalgebraic lattice if and only if the finite extension of ρ\rho is strongly regular; and (3) a binary relation ρ:X⇀Y\rho :X\hspace{0.33em}\rightharpoonup \hspace{0.33em}Y is generalized finitely strongly regular if and only if (Φρ(X),⊆)\left({\Phi }_{\rho }\left(X),\subseteq ) is a quasi-hyperalgebraic lattice if and only if (Φρ(X),⊆)\left({\Phi }_{\rho }\left(X),\subseteq ) equipped with the interval topology is a Priestley space.

Keywords

• strongly regular relation
• finitely strongly regular relation
• generalized finitely strongly regular relation
• priestley spaces
• 06b35
• 54h10
• 06a11