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Quasivariety
In mathematics, a quasivariety is a class of algebraic structures generalizing the notion of variety by allowing equational conditions on the axioms defining the class. __TOC__ Definition A ''trivial algebra'' contains just one element. A quasivariety is a class ''K'' of algebras with a specified signature satisfying any of the following equivalent conditions: # ''K'' is a pseudoelementary class closed under subalgebras and direct products. # ''K'' is the class of all models of a set of quasi-identities, that is, implications of the form s_1 \approx t_1 \land \ldots \land s_n \approx t_n \rightarrow s \approx t, where s, s_1, \ldots, s_n,t, t_1, \ldots, t_n are terms built up from variables using the operation symbols of the specified signature. # ''K'' contains a trivial algebra and is closed under isomorphisms, subalgebras, and reduced products. # ''K'' contains a trivial algebra and is closed under isomorphisms, subalgebras, direct products, and ultraproducts. Examples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoelementary Class
In logic, a pseudoelementary class is a class of structures derived from an elementary class (one definable in first-order logic) by omitting some of its sorts and relations. It is the mathematical logic counterpart of the notion in category theory of (the codomain of) a forgetful functor, and in physics of (hypothesized) hidden variable theories purporting to explain quantum mechanics. Elementary classes are (vacuously) pseudoelementary but the converse is not always true; nevertheless pseudoelementary classes share some of the properties of elementary classes such as being closed under ultraproducts. Definition A pseudoelementary class is a reduct of an elementary class. That is, it is obtained by omitting some of the sorts and relations of a (many-sorted) elementary class. Examples The theory with equality of sets under union and intersection, whose structures are of the form (''W'', ∪, ∩), can be understood naively as the pseudoelementary class formed from the two-s ... [...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, 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] |
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] |
Algebraic Structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of identities (known as ''axioms'') that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called ''scalar multiplication'' between elements of the field (called '' scalars''), and elements of the vector space (called '' vectors''). Abstract algebra is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in universal algebra. Category theory is another formalization that includes also other mathematical structures and functions between structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signature (logic)
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic. Definition Formally, a (single-sorted) signature can be defined as a 4-tuple \sigma = \left(S_, S_, S_, \operatorname\right), where S_ and S_ are disjoint sets not containing any other basic logical symbols, called respectively * '' function symbols'' (examples: +, \times), * ''s'' or '' predicates'' (examples: \,\leq, \, \in), * '' constant symbols'' (examples: 0, 1), and a function \operatorname : S_ \cup S_ \to \N which assigns a natural number called ''arity'' to every function or relation symbol. A function or relation symbol is called n-ary if its arity is n. Some authors define a nullary (0-ary) function symbol as ''constant symbol'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subalgebra
In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of ''all'' algebraic structures. "Subalgebra" can refer to either case. Subalgebras for algebras over a ring or field A subalgebra of an algebra over a commutative ring or field is a vector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field. This notion also applies to most specializations, where the multiplication must satisfy additional properties, e.g. to associative algebras or to Lie algebras. Only for unital algebras is there a stronger notion, of unital subalgebra, for which it is also required that the unit of the subalgebra be the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Product
In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abstraction of these notions in the setting of category theory. Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. The direct sum is a related operation that agrees with the direct product in some but not all cases. Examples * If \R is thought of as the set of real numbers without further structure, the direct product \R \times \R is just the Cartesian product \. * If \R is thought of as the group of real numbers under addition, the direct product \R\times \R still has \ as its underlying set. The difference between this and the preceding examples is that \R \times \R is now a group and so how to add their elements must also be s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model (logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it. Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures of first-order theories with no relation symbols. Model theory has a different scope that encompasses more arbitrary first-order theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic, cf. also Tarski's theory of truth or Tarskian semantics. For a given theory in model theory, a structure is called a model if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a '' semantic model'' when one discusses the notion in the more general setting of mathematical models. Lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-identity
In universal algebra, a quasi-identity is an implication of the form :''s''1 = ''t''1 ∧ … ∧ ''s''''n'' = ''t''''n'' → ''s'' = ''t'' where ''s''1, ..., ''s''''n'', ''t''1, ..., ''t''''n'', ''s'', and ''t'' are terms built up from variables using the operation symbols of the specified signature. A quasi-identity amounts to a conditional equation for which the conditions themselves are equations. Alternatively, it can be seen as a disjunction of inequations and one equation ''s''1 ≠ ''t''1 ∨ ... ∨ ''s''''n'' ≠ ''t''''n'' ∨ ''s'' = ''t''—that is, as a definite Horn clause. A quasi-identity with ''n'' = 0 is an ordinary identity or equation, so quasi-identities are a generalization of identities. See also * Quasivariety In mathematics, a quasivariety is a class of algebraic structures generalizing the notion of variety by allowing equational conditions on the axioms defining the class. __TOC__ Definition A ''trivial algebra'' contains just one element. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Term (logic)
In mathematical logic, a term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact. A first-order term is recursively constructed from constant symbols, variable symbols, and function symbols. An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation. For example, is a term built from the constant 1, the variable , and the binary function symbols and ; it is part of the atomic formula which evaluates to true for each real-numbered value of . Besides in logic, terms play important roles in universal algebra, and rewriting systems. Definition Given a set ''V'' of variable symbols, a set ''C'' of constant symbols and sets ''F''''n'' of ''n''-ary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Product
In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct. Let be a nonempty family of structures of the same signature σ indexed by a set ''I'', and let ''U'' be a proper filter on ''I''. The domain of the reduced product is the quotient of the Cartesian product :\prod_ S_i by a certain equivalence relation ~: two elements (''ai'') and (''bi'') of the Cartesian product are equivalent if :\left\\in U If ''U'' only contains ''I'' as an element, the equivalence relation is trivial, and the reduced product is just the direct product. If ''U'' is an ultrafilter, the reduced product is an ultraproduct. Operations from σ are interpreted on the reduced product by applying the operation pointwise. Relations are interpreted by :R((a^1_i)/,\dots,(a^n_i)/) \iff \\in U. For example, if each structure is a vector space In mathematics and physics, a vector space (also cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
