In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Stone algebra or Stone lattice is a
pseudocomplemented distributive lattice
In mathematics, a distributive lattice is a lattice (order), lattice in which the operations of join and meet distributivity, distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice o ...
''L'' in which any of the following equivalent statements hold for all
*
;
*
;
*
.
They were introduced by and named after
Marshall Harvey Stone
Marshall Harvey Stone (April 8, 1903 – January 9, 1989) was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras.
Biography
Stone was the son of Harlan Fiske Stone, who ...
.
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''.
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
s 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
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
s of a given positive integer is a Stone lattice.
See also
*
De Morgan algebra
__NOTOC__
In mathematics, a De Morgan algebra (named after Augustus De Morgan, a British mathematician and logician) is a structure ''A'' = (A, ∨, ∧, 0, 1, ¬) such that:
* (''A'', ∨, ∧, 0, ...
*
Heyting algebra
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' call ...
References
*
*
*
*
Universal algebra
Lattice theory
Ockham algebras
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