Stone algebra extensions with bounded dense sets

Abstract

Stone algebras have been characterized by Chen and Gratzer in terms of triples \( (B, D, \varphi) \), where D is a distributive lattice with 1, B is a Boolean algebra, and \( \varphi \) is a bounded lattice homomorphism from B into the lattice of filters of D. If D is bounded, the construction of these characterizing triples is much simpler, since the homomorphism \( \varphi \) can be replaced by one from B into D itself. The triple construction leads to natural embeddings of a Stone algebra into ones with bounded dense set. These embeddings correspond to a complete sublattice of the distributive lattice of lattice congruences of S. In addition, the largest embedding is a reflector to the subcategory of Stone algebras with bounded dense sets and morphisms preserving the zero of the dense set.