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∧y)* = x* ∨ y*;
• (x∨y)** = x** ∨ y**;
• x* ∨ x** = 1.

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

The set S(L) ≝ { x** | x ∈ 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

• 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