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Stone Algebra
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 x, y \in L: * (x\wedge y)^* = x^*\vee y^*; * (x\vee y)^ = x^\vee y^; * x^* \vee x^ = 1. They were introduced by and named after Marshall Harvey Stone. The set S(L) \stackrel \ 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 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 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 ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Pseudocomplemented
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 greatest element x^*\in L with the property that x\wedge x^*=0. More formally, x^* = \max\. The lattice ''L'' itself is called a pseudocomplemented lattice if every element of ''L'' is pseudocomplemented. Every pseudocomplemented lattice is necessarily bounded, i.e. it has a 1 as well. Since the pseudocomplement is unique by definition (if it exists), a pseudocomplemented lattice can be endowed with a unary operation * mapping every element to its pseudocomplement; this structure is sometimes called a ''p''-algebra. However this latter term may have other meanings in other areas of mathematics. Properties In a ''p''-algebra ''L'', for all x, y \in L: * The map x \mapsto x^* is antitone. In particular, 0^* = 1 and 1^* = 0. * The map x \mapsto x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
