Stone lattice

A pseudo-complemented distributive lattice $ L $( see Lattice with complements) in which $ a ^ \star + a ^ {\star\star} = 1 $ for all $ a \in L $. A pseudo-complemented distributive lattice $ L $ is a Stone lattice if and only if the join of any two of its minimal prime ideals is the whole of $ L $( the Grätzer–Schmidt theorem, [3]).

A Stone lattice, considered as a universal algebra with the basic operations $ \langle \lor , \wedge , {} ^ \star , 0, 1\rangle $, is called a Stone algebra. Every Stone algebra is a subdirect product of two-element and three-element chains. In a pseudo-complemented lattice, an element $ x $ is said to be dense if $ x ^ \star = 0 $. The centre $ C( L) $ of a Stone lattice $ L $( cf. Centre of a partially ordered set) is a Boolean algebra, while the set $ D( L) $ of all its dense elements is a distributive lattice with a unit. Moreover, there is a homomorphism $ \phi ^ {L} $ from $ C( L) $ into the lattice $ F( D( L)) $ of filters of $ D( L) $, defined by

$$ a \phi ^ {L} = \ \{ {x } : {x \in D( L), x \geq a ^ \star } \} , $$

which preserves 0 and 1.

The triplet $ \langle C( L), D( L), \phi ^ {L} \rangle $ is said to be associated with the Stone algebra $ L $. Homomorphisms and isomorphisms of triplets are defined naturally. Any triplet $ \langle C, D, \phi \rangle $, where $ C $ is a Boolean algebra, $ D $ is a distributive lattice with a $ 1 $ and $ \phi : C \rightarrow F( D) $ is a homomorphism preserving 0 and 1, is isomorphic to the triplet associated with some Stone algebra. Stone algebras are isomorphic if and only if their associated triplets are isomorphic (the Chen–Grätzer theorem, [2]).

References

[1]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[2]	C.C. Chen, G. Grätzer, "Stone lattices I-II" Canad. J. Math. , 21 : 4 (1969) pp. 884–903
[3]	G. Grätzer, E.T. Schmidt, "On a problem of M.H. Stone" Acta Math. Acad. Sci. Hung. , 8 : 3–4 (1957) pp. 455–460
[4]	T.S. Fofanova, "General theory of lattices" , Ordered sets and lattices , 3 , Saratov (1975) pp. 22–40 (In Russian)

Comments

Stone lattices occur, in particular, as the open-set lattices of extremally-disconnected spaces (see Extremally-disconnected space), and are so named in honour of M.H. Stone's investigation of such spaces [a1]. If $ L $ is the lattice of all open sets of a compact extremally-disconnected space $ X $, then $ C( L) $ is a complete Boolean algebra, and $ X $ is its Stone space; thus, in this case $ L $ is entirely determined by $ C( L) $.

References

[a1]	M.H. Stone, "Algebraic characterization of special Boolean rings" Fund. Math. , 29 (1937) pp. 223–303
[a2]	G. Grätzer, "General lattice theory" , Birkhäuser (1978) (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978)