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Ockham Algebras
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 algebras, and 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 ...s. References * (pd availablefrom GDZ) * * * {{algebra-stub Algebraic logic * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Lattice
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These ''lattice-like'' structures all admit ... [...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] |
Duality (order Theory)
In the mathematical area of order theory, every partially ordered set ''P'' gives rise to a dual (or opposite) partially ordered set which is often denoted by ''P''op or ''P''''d''. This dual order ''P''op is defined to be the same set, but with the inverse order, i.e. ''x'' ≤ ''y'' holds in ''P''op if and only if ''y'' ≤ ''x'' holds in ''P''. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for ''P'' upside down, will indeed yield a partially ordered set. In a broader sense, two partially ordered sets are also said to be duals if they are dually isomorphic, i.e. if one poset is order isomorphic to the dual of the other. The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the Duality Principle for ordered sets: : If a given statement is valid for all partially ordered sets, then its dual statement, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Of Ockham
William of Ockham, OFM (; also Occam, from la, Gulielmus Occamus; 1287 – 10 April 1347) was an English Franciscan friar, scholastic philosopher, apologist, and Catholic theologian, who is believed to have been born in Ockham, a small village in Surrey. He is considered to be one of the major figures of medieval thought and was at the centre of the major intellectual and political controversies of the 14th century. He is commonly known for Occam's razor, the methodological principle that bears his name, and also produced significant works on logic, physics and theology. William is remembered in the Church of England with a commemoration on 10 April. Life William of Ockham was born in Ockham, Surrey in 1287. He received his elementary education in the London House of the Greyfriars. It is believed that he then studied theology at the University of OxfordSpade, Paul Vincent (ed.). ''The Cambridge Companion to Ockham''. Cambridge University Press, 1999, p. 20.He has long be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variety (universal Algebra)
In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to #Birkhoff's_theorem, Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphism, homomorphic images, subalgebras and Direct product#Direct product in universal algebra, (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a Category (mathematics), category; these are usually called ''finitary algebraic categories''. A ''covariety'' is the class of all F-coalgebra, coalgebraic structures of a given signature. Terminology A variety of algebras should not be confused with an algebraic variety, which means a set of solutions to a system of polynomial eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (''and'') denoted as ∧, disjunction (''or'') denoted as ∨, and the negation (''not'') denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction and division. So Boolean algebra is a formal way of describing logical operations, in the same way that elementary algebra describes numerical operations. Boolean algebra was introduced by George Boole in his first book ''The Mathematical Analysis of Logic'' (1847), and set forth more fully in his '' An Investigation of the Laws of Thought'' (1854). According to Huntington, the term "Boolean algebra" wa ... [...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] |
Kleene Algebra (with Involution)
__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] |
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] |
Aequationes Mathematicae
''Aequationes Mathematicae'' is a mathematical journal. It is primarily devoted to functional equations, but also publishes papers in dynamical systems, combinatorics, and geometry. As well as publishing regular journal submissions on these topics, it also regularly reports on international symposia on functional equations and produces bibliographies on the subject. János Aczél founded the journal in 1968 at the University of Waterloo, in part because of the long publication delays of up to four years in other journals at the time of its founding. It is currently published by Springer Science+Business Media, with Zsolt Páles of the University of Debrecen as its editor in chief. János Aczél remains its honorary editor in chief. it was listed as a second-quartile mathematics journal by SCImago Journal Rank The SCImago Journal Rank (SJR) indicator is a measure of the prestige of scholarly journals that accounts for both the number of citations received by a journal and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Center For Retrospective Digitization
The Center for Retrospective Digitization in Göttingen (german: Göttinger DigitalisierungsZentrum, GDZ) is an online system for archiving academic journals maintained by the University of Göttingen. See also *JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... * List of retrodigitized Mathematics Journals and Monograph References External linksOfficial website (German only) German digital libraries Academic publishing Göttingen {{database-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Logic
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems) and connected problems like representation and duality. Well known results like the representation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic . Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator . Calculus of relations A homogeneous binary relation is found in the power set of ''X'' × ''X'' for some set ''X'', while a heterogeneous relation is found in the power set of ''X'' × ''Y'', where ''X'' ≠ ''Y''. Whether a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
