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 $x,y\in L:$

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

They were introduced by , and named after Marshall Harvey Stone.

The set $S(L)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\{x^{\ast }\mid x\in L\}$ 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

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
• Grätzer, George (1971), Lattice theory. First concepts and distributive lattices, W. H. Freeman and Co., ISBN 978-0-486-47173-0, MR 0321817

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