Stone lattice

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

A Stone lattice, considered as a universal algebra with the basic operations , 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 is said to be dense if . The centre of a Stone lattice (cf. Centre of a partially ordered set) is a Boolean algebra, while the set of all its dense elements is a distributive lattice with a unit. Moreover, there is a homomorphism from into the lattice of filters of , defined by

which preserves 0 and 1.

The triplet is said to be associated with the Stone algebra . Homomorphisms and isomorphisms of triplets are defined naturally. Any triplet , where is a Boolean algebra, is a distributive lattice with a and 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 is the lattice of all open sets of a compact extremally-disconnected space , then is a complete Boolean algebra, and is its Stone space; thus, in this case is entirely determined by .

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)