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
- ;
- ;
- .
They were introduced by {{harvtxt|Grätzer|Schmidt|1957}} and named after Marshall Harvey Stone.
The set 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.{{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}}
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
References
{{Reflist}}
- {{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 | last2=Schmidt | first2=E. T. | title=On a problem of M. H. Stone | doi=10.1007/BF02020328 | doi-access=free | mr=0092763 | year=1957 | journal=Acta Mathematica Academiae Scientiarum Hungaricae | issn=0001-5954 | volume=8 | issue=3–4 | pages=455–460}}
- {{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}}
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