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Monadic Boolean Algebra
In abstract algebra, a monadic Boolean algebra is an algebraic structure ''A'' with signature : of type ⟨2,2,1,0,0,1⟩, where ⟨''A'', ·, +, ', 0, 1⟩ is a Boolean algebra. The monadic/unary operator ∃ denotes the existential quantifier, which satisfies the identities (using the received prefix notation for ∃): * * * * is the ''existential closure'' of ''x''. Dual to ∃ is the unary operator ∀, the universal quantifier, defined as . A monadic Boolean algebra has a dual definition and notation that take ∀ as primitive and ∃ as defined, so that . (Compare this with the definition of the dual Boolean algebra.) Hence, with this notation, an algebra ''A'' has signature , with ⟨''A'', ·, +, ', 0, 1⟩ a Boolean algebra, as before. Moreover, ∀ satisfies the following dualized version of the above identities: # # # # . is the ''universal closure'' of ''x''. Discussion Monadic Boolean algebras have an important connection to topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called ''finitary algebraic categories''. A ''covariety'' is the class of all 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 equations. They are formally quite distinct and their theories have little in common. The term "variety of algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Łukasiewicz–Moisil Algebra
Łukasiewicz–Moisil algebras (LM''n'' algebras) were introduced in the 1940s by Grigore Moisil (initially under the name of Łukasiewicz algebras) in the hope of giving Algebraic semantics (mathematical logic), algebraic semantics for the ''n''-valued Łukasiewicz logic. However, in 1956 Alan Rose discovered that for ''n'' ≥ 5, the Łukasiewicz–Moisil algebra does not model (mathematical logic), model the Łukasiewicz logic. A faithful model for the ℵ0-valued (infinitely-many-valued) Łukasiewicz–Tarski logic was provided by C. C. Chang's MV-algebra, introduced in 1958. For the axiomatically more complicated (finite) ''n''-valued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MV-algebra#MVn-algebras, MV''n''-algebras. MV''n''-algebras are a subclass of LM''n''-algebras, and the inclusion is strict for ''n'' ≥ 5.Iorgulescu, A.: Connections between MV''n''-algebras and ''n''-valued Łukasiewicz-Moisil algebras—I. Discrete Math. 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Kuratowski Closure Axioms
In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a Set (mathematics), set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski, and the idea was further studied by mathematicians such as Wacław Sierpiński and Antonio Monteiro (mathematician), António Monteiro, among others. A similar set of axioms can be used to define a topological structure using only the dual notion of Interior (topology)#Interior operator, interior operator. Definition Kuratowski closure operators and weakenings Let X be an arbitrary set and \wp(X) its power set. A Kuratowski closure operator is a unary operation \mathbf:\wp(X) \to \wp(X) with the following properties: A consequence of \mathbf preserving binary unions is the following condition: In fact if we rewrite the equality in [K4] as an inclusion, giving the weaker axiom [K4''] ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Interior Algebra
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras. Definition An interior algebra is an algebraic structure with the signature :⟨''S'', ·, +, ′, 0, 1, I⟩ where :⟨''S'', ·, +, ′, 0, 1⟩ is a Boolean algebra and postfix I designates a unary operator, the interior operator, satisfying the identities: # ''x''I ≤ ''x'' # ''x''II = ''x''I # (''xy'')I = ''x''I''y''I # 1I = 1 ''x''I is called the interior of ''x''. The dual of the interior operator is the closure operator C defined by ''x''C = ((''x''′)I)′. ''x''C is called the closure of ''x''. By the principle of duality, the closure operator satisfies the identities: # ''x''C ≥ ''x'' # ''x''CC = ''x''C # (''x'' + ''y'')C = ''x''C + ''y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cylindric Algebra
In mathematics, the notion of cylindric algebra, developed by Alfred Tarski, arises naturally in the Algebraic logic, algebraization of first-order logic with equality. This is comparable to the role Boolean algebra (structure), Boolean algebras play for propositional logic. Cylindric algebras are Boolean algebras equipped with additional cylindrification operations that model Quantification (logic), quantification and equality (mathematics), equality. They differ from polyadic algebras in that the latter do not model equality. The cylindric algebra should not be confused with the measure theory, measure theoretic concept ''cylindrical algebra'' that arises in the study of cylinder set measures and the cylindrical σ-algebra. Definition of a cylindric algebra A cylindric algebra of dimension \alpha (where \alpha is any ordinal number) is an algebraic structure (A,+,\cdot,-,0,1,c_\kappa,d_)_ such that (A,+,\cdot,-,0,1) is a Boolean algebra (structure), Boolean algebra, c_\kappa a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Clopen Set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counterintuitive, as the common meanings of and are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open closed, and therefore clopen. As described by topologist James Munkres, unlike a door, "a set can be open, or closed, or both, or neither!" emphasizing that the meaning of "open"/"closed" for is unrelated to their meaning for (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as " door spaces" their name. Examples In any topological space X, the empty set and the whole space X are both clopen. Now consider the space X which consists of the union of the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Synonym
A synonym is a word, morpheme, or phrase that means precisely or nearly the same as another word, morpheme, or phrase in a given language. For example, in the English language, the words ''begin'', ''start'', ''commence'', and ''initiate'' are all synonyms of one another: they are ''synonymous''. The standard test for synonymy is substitution: one form can be replaced by another in a sentence without changing its meaning. Words may often be synonymous in only one particular sense: for example, ''long'' and ''extended'' in the context ''long time'' or ''extended time'' are synonymous, but ''long'' cannot be used in the phrase ''extended family''. Synonyms with exactly the same meaning share a seme or denotational sememe, whereas those with inexactly similar meanings share a broader denotational or connotational sememe and thus overlap within a semantic field. The former are sometimes called cognitive synonyms and the latter, near-synonyms, plesionyms or poecilonyms. Lexic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Modal Logic
Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality, causation. For instance, in epistemic modal logic, the well-formed_formula, formula \Box P can be used to represent the statement that P is known. In deontic modal logic, that same formula can represent that P is a moral obligation. Modal logic considers the inferences that modal statements give rise to. For instance, most epistemic modal logics treat the formula \Box P \rightarrow P as a Tautology_(logic), tautology, representing the principle that only true statements can count as knowledge. However, this formula is not a tautology in deontic modal logic, since what ought to be true can be false. Modal logics are formal systems that include unary operation, unary operators such as \Diamond and \Box, representing possibility and necessi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Paul Halmos
Paul Richard Halmos (; 3 March 1916 – 2 October 2006) was a Kingdom of Hungary, Hungarian-born United States, American mathematician and probabilist who made fundamental advances in the areas of mathematical logic, probability theory, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor. He has been described as one of The Martians (scientists), The Martians. Early life and education Born in the Kingdom of Hungary into a History of the Jews in Hungary, Jewish family, Halmos immigrated to the United States at age 13. He obtained his B.A. from the University of Illinois at Urbana-Champaign, University of Illinois, majoring in mathematics while also fulfilling the requirements for a degree in philosophy. He obtained the degree after only three years, and was 19 years old when he graduated. He then began a Ph.D. in philosophy, still at the Champaign–Urbana campus. However, after failin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
First-order Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Polyadic Algebra
Polyadic algebras (more recently called Halmos algebras) are algebraic structures introduced by Paul Halmos. They are related to first-order logic analogous to the relationship between Boolean algebras and propositional logic (see Lindenbaum–Tarski algebra). There are other ways to relate first-order logic to algebra, including Tarski's cylindric algebras (when equality is part of the logic) and Lawvere's functorial semantics (a categorical approach). References Further reading *Paul Halmos, ''Algebraic Logic'', Chelsea Publishing The Chelsea Publishing Company was a publisher of mathematical books, based in New York City New York, often called New York City (NYC), is the most populous city in the United States, located at the southern tip of New York State on on ..., New York (1962) Algebraic logic {{mathlogic-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
