Stone algebra

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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 ${\displaystyle x,y\in L:}$

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

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

The set ${\displaystyle S(L){\stackrel {\mathrm {def} }{=}}\{x^{*}\mid x\in L\}}$ 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

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