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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 relationρ:X⇀Y\rho :X\hspace{0.33em}\rightharpoonup \hspace{0.33em}Yis 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 relationρ:X⇀Y\rho :X\hspace{0.33em}\rightharpoonup \hspace{0.33em}Yis 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 ofρ\rhois strongly regular; and (3) a binary relationρ:X⇀Y\rho :X\hspace{0.33em}\rightharpoonup \hspace{0.33em}Yis 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.

Relational representations of algebraic lattices and their applications