Stone algebra

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 ${\displaystyle x,y\in L:}$^[1]

• ${\displaystyle (x\wedge y)^{*}=x^{*}\vee y^{*}}$;
• ${\displaystyle (x\vee y)^{**}=x^{**}\vee y^{**}}$;
• ${\displaystyle x^{*}\vee x^{**}=1}$.

They were introduced by Grätzer & Schmidt (1957) and named after Marshall Harvey Stone.

The set ${\displaystyle S(L){\stackrel {\mathrm {d} ef}{=}}\{x^{*}\mid x\in L\}}$ 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.^[1]

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