In mathematics, a Stone algebra or Stone lattice is a pseudocomplemented distributive lattice L in which any of the following equivalent statements hold for all x , y ∈ L : {\displaystyle x,y\in L:}
( x ∧ y ) ∗ = x ∗ ∨ y ∗ {\displaystyle (x\wedge y)^{*}=x^{*}\vee y^{*}} ;
( x ∨ y ) ∗ ∗ = x ∗ ∗ ∨ y ∗ ∗ {\displaystyle (x\vee y)^{**}=x^{**}\vee y^{**}} ;
x ∗ ∨ x ∗ ∗ = 1 {\displaystyle x^{*}\vee x^{**}=1} . They were introduced by Grätzer & Schmidt (1957), and named after Marshall Harvey Stone. The set S ( L ) = d e f { x ∗ ∣ x ∈ L } {\displaystyle S(L){\stackrel {\mathrm {def} }{=}}\{x^{*}\mid x\in L\}} is called the skeleton of L. Then L is a Stone algebra if and only if its skeleton S(L) is a sublattice of L. Boolean algebras are Stone algebras, and Stone algebras are Ockham algebras.
Examples The open-set lattice of an extremally disconnected space is a Stone algebra. The lattice of positive divisors of a given positive integer is a Stone lattice.
See also De Morgan algebra Heyting algebra
References
Further reading Balbes, Raymond (1970), "A survey of Stone algebras", Proceedings of the Conference on Universal Algebra (Queen's Univ., Kingston, Ont., 1969), Kingston, Ont.: Queen's Univ., pp. 148–170, MR 0260638 Fofanova, T.S. (2001) [1994], "Stone lattice", Encyclopedia of Mathematics, EMS Press Grätzer, George (1971), Lattice theory. First concepts and distributive lattices, W. H. Freeman and Co., ISBN 978-0-486-47173-0, MR 0321817