In
mathematics, a Stone algebra, or Stone lattice, is a
pseudo-complemented In mathematics, particularly in order theory, a pseudocomplement is one generalization of the notion of complement. In a lattice ''L'' with bottom element 0, an element ''x'' ∈ ''L'' is said to have a ''pseudocomplement'' if there exists a gr ...
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.
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 ...
s are Stone algebras, and Stone algebras are
Ockham algebra In mathematics, an Ockham algebra is a bounded distributive lattice with a dual endomorphism, that is, an operation ~ satisfying ~(''x'' ∧ ''y'') = ~''x'' ∨ ~''y'', ~(''x'' ∨ ''y'') = ~''x'' ∧ ~''y'', ~0 = 1, ~1 = 0. They were introduced b ...
s.
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 divisible by ...
s of a given positive integer is a Stone lattice.
See also
*
De Morgan algebra
*
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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