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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 <math>x, y \in L:</math><ref name="Blyth" /> * <math>(x\wedge y)^* = x^*\vee y^*</math>; * <math>(x\vee y)^{**} = x^{**}\vee y^{**}</math>; * <math>x^* \vee x^{**} = 1</math>.
They were introduced by {{harvtxt|Grätzer|Schmidt|1957}},<ref>{{Citation |last1=Grätzer |first1=George |title=On a problem of M. H. Stone |journal=Acta Mathematica Academiae Scientiarum Hungaricae |volume=8 |issue=3–4 |pages=455–460 |year=1957 |doi=10.1007/BF02020328 |issn=0001-5954 |mr=0092763 |last2=Schmidt |first2=E. T. |doi-access=free}}</ref> and named after Marshall Harvey Stone.
The set <math>S(L) \stackrel{\mathrm{def}}{=} \{ x^* \mid x\in L \}</math> 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''.<ref name="Blyth">{{cite book|author=T.S. Blyth|title=Lattices and Ordered Algebraic Structures|year=2006|publisher=Springer Science & Business Media|isbn=978-1-84628-127-3|at=Chapter 7. Pseudocomplementation; Stone and Heyting algebras. pp. 103–119}}</ref>
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== {{Reflist}}
== Further reading == *{{Citation | last1=Balbes | first1=Raymond | title=Proceedings of the Conference on Universal Algebra (Queen's Univ., Kingston, Ont., 1969) | url=https://books.google.com/books?id=_bsrAAAAYAAJ | publisher=Queen's Univ. | location=Kingston, Ont. | mr=0260638 | year=1970 | chapter=A survey of Stone algebras | pages=148–170}} *{{eom|id=s/s090350|title=Stone lattice|first=T.S. |last=Fofanova}} *{{Citation | last1=Grätzer | first1=George | title=Lattice theory. First concepts and distributive lattices | url=https://books.google.com/books?id=R6adPQAACAAJ | publisher=W. H. Freeman and Co. | isbn=978-0-486-47173-0 | mr=0321817 | year=1971}}
Category:Universal algebra Category:Lattice theory Category:Ockham algebras
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