Stone algebra extensions with bounded dense sets (Q1272081)

Stone algebras were characterized by Chen and Grätzer in terms of triplets \((B,D,\varphi),\) where \(B\) is a Boolean algebra, \(D\) is a distributive lattice with 1 and \(\varphi \) is a bounded lattice homomorphism from \(B\) into the lattice of filters of \(D.\) If \(D\) is bounded, the construction can be simplified since the homomorphism \(\varphi \) can be replaced by one from \(B\) into \(D\) itself. The triple construction leads to a natural embedding of a Stone algebra \(S\) into \(\overline S\) with bounded dense set. More precisely, a bounded dense extension of a Stone algebra \(S\) is a Stone algebra \(T\) and a Stone algebra monic \(\gamma \: S\to T\) with (1) the set \(E\) of dense elements of \(T\) has a smallest element, (2) \(T\) is generated as a Stone algebra by \(\gamma (S)\cup \{0_E\}.\) The set of all bounded extensions is ordered and the smallest bounded dense extension \(\overline S\) of \(S\) in this order is constructed. It is shown that in fact this ordered set is a complete bounded distributive lattice and that the category of Stone algebras with bounded dense sets and strong homomorphisms is a reflexive subcategory of Stone algebras with Stone algebra homomorphisms.