In
mathematics, a Stone algebra, or Stone lattice, is a
pseudo-complemented distributive lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set ...
such that ''a''* ∨ ''a''** = 1. 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 w ...
.
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 variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
s are Stone algebras, and Stone algebras are
Ockham algebras.
Examples:
* The open-set lattice of an
extremally disconnected space In mathematics, an extremally disconnected space is a topological space in which the closure of every open set is open. (The term "extremally disconnected" is correct, even though the word "extremally" does not appear in most dictionaries, and is so ...
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 divisible by ...
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,&nbs ...
*
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'' of '' ...
References
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Universal algebra
Lattice theory
Ockham algebras
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