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C-minimal Theory
In model theory, a branch of mathematical logic, a C-minimal theory is a theory that is "minimal" with respect to a ternary relation ''C'' with certain properties. Algebraically closed fields with a (Krull) valuation are perhaps the most important example. This notion was defined in analogy to the o-minimal theories, which are "minimal" (in the same sense) with respect to a linear order. Definition A ''C''-relation is a ternary relation that satisfies the following axioms. # \forall xyz\, C(x;y,z)\rightarrow C(x;z,y) # \forall xyz\, C(x;y,z)\rightarrow\neg C(y;x,z) # \forall xyzw\, C(x;y,z)\rightarrow (C(w;y,z)\vee C(x;w,z)) # \forall xy\, x\neq y \rightarrow \exists z\neq y\, C(x;y,z) A C-minimal structure is a structure ''M'', in a signature containing the symbol ''C'', such that ''C'' satisfies the above axioms and every set of elements of ''M'' that is definable with parameters in ''M'' is a Boolean combination of instances of ''C'', i.e. of formulas of the form , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be definable set, defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triadic Relation
In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relation is formally defined as a set of ''pairs'', i.e. a subset of the Cartesian product of some sets ''A'' and ''B'', so a ternary relation is a set of triples, forming a subset of the Cartesian product of three sets ''A'', ''B'' and ''C''. An example of a ternary relation in elementary geometry can be given on triples of points, where a triple is in the relation if the three points are collinear. Another geometric example can be obtained by considering triples consisting of two points and a line, where a triple is in the ternary relation if the two points determine (are incident with) the line. Examples Binary functions A function in two variables, mapping two values from sets ''A'' and ''B'', respectively, to a value in ''C'' ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed Field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra holds for it. Every field K is contained in an algebraically closed field C, and the roots in C of the polynomials with coefficients in K form an algebraically closed field called an algebraic closure of K. Given two algebraic closures of K there are isomorphisms between them that fix the elements of K. Algebraically closed fields appear in the following chain of class inclusions: Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation x^2+1=0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation (algebra)
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field. Definition One starts with the following objects: *a field and its multiplicative group ''K''×, *an abelian totally ordered group . The ordering and group law on are extended to the set by the rules * for all ∈ , * for all ∈ . Then a valuation of is any map : that satisfies the following properties for all ''a'', ''b'' in ''K'': * if and only if , *, *, with equality if ''v' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
O-minimal Theory
In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) that is totally ordered by < is called an o-minimal structure if and only if every definable subset ''X'' ⊆ ''M'' (with parameters taken from ''M'') is a finite union of intervals and points. O-minimality can be regarded as a weak form of . A structure ''M'' is o-minimal if and only if every |
Structure (mathematical 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. Log ... [...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] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
