|
Stone Algebra
In mathematics, a Stone algebra, or Stone lattice, is a pseudo-complemented distributive lattice such that ''a''* ∨ ''a''** = 1. They were introduced by and named after Marshall Harvey Stone. 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 operation ''a'' → ''b'' of '' ... References * * * * Universal algebra Lattice theory Ockham algebras {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 greatest element ''x''* ∈ ''L'' with the property that ''x'' ∧ ''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'' ↦ ''x''* is antitone. In particular, 0* = 1 and 1* = 0. * The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets. Definition As in the case of arbitrary lattices, one can choose to consider a distributive lattice ''L'' either as a structure of order theory or of universal algebra. Both views and their mutual correspondence are discussed in the article on lattices. In the present situation, the algebraic description appears to be more convenient. A lattice (''L'',∨,∧) is distributive if the following additional identity holds for all ''x'', ''y'', and ''z'' in ''L'': : ''x'' ∧ (''y'' ∨ ''z'') = (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z''). Viewing lattices as partially ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 was the Chief Justice of the United States in 1941–1946. Marshall Stone's family expected him to become a lawyer like his father, but he became enamored of mathematics while he was a Harvard University undergraduate. He completed a Harvard PhD in 1926, with a thesis on differential equations that was supervised by George David Birkhoff. Between 1925 and 1937, he taught at Harvard, Yale University, and Columbia University. Stone was promoted to a full professor at Harvard in 1937. During World War II, Stone did classified research as part of the "Office of Naval Operations" and the "Office of the Chief of Staff" of the United States Department of War. In 1946, he became the chairman of the Mathematics Department at the University of Chic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. __TOC__ History The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 by , and were named after William of Ockham by . Ockham algebras form a variety. Examples of Ockham algebras include Boolean algebras, De Morgan algebras, Kleene algebra In mathematics, a Kleene algebra ( ; named after Stephen Cole Kleene) is an idempotent (and thus partially ordered) semiring endowed with a closure operator. It generalizes the operations known from regular expressions. Definition Various ine ...s, and Stone algebras. References * (pd availablefrom GDZ) * * * {{algebra-stub Algebraic logic * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 sometimes mistaken by spellcheckers for the homophone ''extremely disconnected''.) An extremally disconnected space that is also compact and Hausdorff is sometimes called a Stonean space. This is not the same as a Stone space, which is a totally disconnected compact Hausdorff space. Every Stonean space is a Stone space, but not vice versa. In the duality between Stone spaces and Boolean algebras, the Stonean spaces correspond to the complete Boolean algebras. An extremally disconnected first-countable collectionwise Hausdorff space must be discrete. In particular, for metric spaces, the property of being extremally disconnected (the closure of every open set is open) is equivalent to the property of being discrete (every set is op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer is divisible by a nonzero integer if there exists an integer such that n=km. This is written as :m\mid n. Other ways of saying the same thing are that divides , is a divisor of , is a factor of , and is a multiple of . If does not divide , then the notation is m\not\mid n. Usually, is required to be nonzero, but is allowed to be zero. With this convention, m \mid 0 for every nonzero integer . Some definitions omit the requirement that m be nonzero. General Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, 1) is a bounded distributive lattice, and * ¬ is a De Morgan involution: ¬(''x'' ∧ ''y'') = ¬''x'' ∨ ¬''y'' and ¬¬''x'' = ''x''. (i.e. an involution that additionally satisfies De Morgan's laws) In a De Morgan algebra, the laws * ¬''x'' ∨ ''x'' = 1 (law of the excluded middle), and * ¬''x'' ∧ ''x'' = 0 (law of noncontradiction) do not always hold. In the presence of the De Morgan laws, either law implies the other, and an algebra which satisfies them becomes a Boolean algebra. Remark: It follows that ¬(x ∨ y) = ¬x ∧ ¬y, ¬1 = 0 and ¬0 = 1 (e.g. ¬1 = ¬1 ∨ 0 = ¬1 ∨ ¬¬0 = ¬(1 ∧ ¬0) = ¬¬0 = 0). Thus ¬ is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ''implication'' such that (''c'' ∧ ''a'') ≤ ''b'' is equivalent to ''c'' ≤ (''a'' → ''b''). From a logical standpoint, ''A'' → ''B'' is by this definition the weakest proposition for which modus ponens, the inference rule ''A'' → ''B'', ''A'' ⊢ ''B'', is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by to formalize intuitionistic logic. As lattices, Heyting algebras are distributive. Every Boolean algebra is a Heyting algebra when ''a'' → ''b'' is defined as ¬''a'' ∨ ''b'', as is every complete distributive lattice satisfying a one-sided infinite distributive law when ''a'' → ''b'' is taken to be the supremum of the set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Mathematica Hungarica
'' Acta Mathematica Hungarica'' is a peer-reviewed mathematics journal of the Hungarian Academy of Sciences, published by Akadémiai Kiadó and Springer Science+Business Media. The journal was established in 1950 and publishes articles on mathematics related to work by Hungarian mathematicians. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.39, and its 2015 impact factor was 0.469. The editor-in-chief is Imre Bárány, honorary editor is Ákos Császár, the editors are the mathematician members of the Hungarian Academy of Sciences. Abstracting and indexing According to the ''Journal Citation Reports'', the journal had a 2020 impact factor of 0.623. This journal is indexed by the following services: * Science Citation Index * Journal Citation Reports/Science Edition * Scopus * Mathematical Reviews * Zentralblatt Math zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study. Basic idea In universal algebra, an algebra (or algebraic structure) is a set ''A'' together with a collection of operations on ''A''. An ''n''- ary operation on ''A'' is a function that takes ''n'' elements of ''A'' and returns a single element of ''A''. Thus, a 0-ary operation (or ''nullary operation'') can be represented simply as an element of ''A'', or a '' constant'', often denoted by a letter like ''a''. A 1-ary operation (or ''unary operation'') is simply a function from ''A'' to ''A'', often denoted by a symbol placed in front of its argument, like ~''x''. A 2-ary operation (or ''binary operation'') is often denoted by a symbol placed between its argum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
