Stone algebra

This article includes inline citations, but they are not properly formatted. Please improve this article by correcting them. (May 2024) (Learn how and when to remove this message)

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.

• De Morgan algebra
• Heyting algebra

• 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; Schmidt, E. T. (1957), "On a problem of M. H. Stone", Acta Mathematica Academiae Scientiarum Hungaricae, 8 (3–4): 455–460, doi:10.1007/BF02020328, ISSN 0001-5954, MR 0092763
• Grätzer, George (1971), Lattice theory. First concepts and distributive lattices, W. H. Freeman and Co., ISBN 978-0-486-47173-0, MR 0321817

This algebra-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e