Balanced d-lattices are complemented

Abstract

We show that all balanced d-lattices must be complemented, answering a question of Chajda and Eigenthaler. (A bounded lattice is balanced if any two congruences agree on their 1-classes iff they agree on their 0-classes.) Our main tool is the characterization of d-lattices (a class of bounded lattices including the bounded distributive lattices, originally defined by a property of their compact congruences) as exactly those lattices in which all maximal filters/ideals are prime.