Question

Prove that every bounded chain is a stone algebra

Answer 1

Stone algebras been characterized by Chen and Grätzer in terms of triples (B,D,φ) where (D) is a distributive lattice and B is Boolean algebra and φ is a bounded lattice homomorphism from B into lattice of filters of D.  

D is surrounded by construction of these characterizing triples and is very simpler from homomorphism φ can be replaced from B into D itself.  

Triple construction leads to natural embeddings of Stone algebra into ones with bounded dense set.  

These embeddings corresponds to complete sublattice of distributive lattice of lattice congruences of S.

The largest embedding is the subcategory of Stone algebras with bounded sets and morphisms storing the zero of dense set.

Answer 2

Answer:

every bounded chain is a stone algebra