Valuation (mathematics)

Informally, a valuation is an assignment of particular values to the variables in a mathematical statement or equation.

In logic and model theory, a valuation is either (i) an assignment of truth values to every atomic sentence, provided each element of the domain has a name in the case of first-order or higher languages, or (ii) a function from non-logical vocabulary to their corresponding objects defined on the domain (e.g. a function taking relation and function symbols to relations and functions defined on the domain, and constants to elements in the domain).

In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a measure of size or multiplicity. They generalize 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 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.

In measure theory or at least in the approach to it though domain theory, a valuation is a map from the class of open sets of a topological space to the set positive real numbers including infinity. It is a concept closely related to that of a measure and as such it finds applications measure theory, probability theory and also in theoretical computer science.

Logic/Model theory definition

The starting point of the discourse is a given formal language
where is its alphabet, a set of transformation rules on , and the closure of under the elements of — i.e. the set of (well-formed) formulas. Given an abstract algebra with three binary operations and one unary operation, which can be the algebra of formulas of the language if the language itself is of order or , i.e.



with properly defined logical disjunction , logical conjunction , logical implication and logical negation , a valuation is any map



where is the set of propositional variables of the language . Thus, a valuation maps propositional variables to algebraic formulas in : details on logic concepts can be found in .

Algebraic definition

To define the algebraic concept of valuation, the following objects are needed: and also an element such that


Then a valuation is any map



which satisfies the following properties



Note that some authors use the term exponential valuation rather than "valuation". In this case the term "valuation" means "absolute value".

For valuations used in geometric applications, the first property implies that any non-empty germ of an analytic variety near a point contains that point. The second property assert that any valuation is a group homomorphism, while the third property is a translation of the triangle inequality from metric spaces to ordered groups.

It is possible to give a dual definition of the same concept: if, instead of , an element is given such that


then a valuation is any map



satisfying the following properties (written using the multiplicative notation for group operation)



A valuation is commonly required to be surjective, since many arguments used in ordinary mathematical research involving those objects use preimages of unspecified elements of the ordered group contained in its codomain. Also, the first definition of valuation given is more frequently encountered in ordinary mathematical research, thus it is the only one used in the following considerations and examples: then, in what follows, is the identity element the ordered group, or the zero element of the ring considered. See for further details.

Equivalence of valuations

Two valuations are said to be equivalent if they have the same domain, codomain and are proportional i.e. they differ by a fixed element belonging to the ordered group in their codomain: using a symbolic notation



Proportionality in this sense is an equivalence relation:
,
  • It is transitive since, given three valuation such that is equivalent to which is in turn equivalent to , then


Every equivalence class of valuations over a field with respect to this equivalence relation is called a place. Ostrowski's theorem gives a complete classification of places of the field of rational numbers : these are precisely equivalence classes of valuations for the p-adic completions of .

Dedekind valuation

A Dedekind valuation is a valuation for which the ordered abelian group in its codomain is the additive group of the integers, i.e.
Dedekind valuations are also known under the name of discrete valuations, even if some authors consider a discrete valuation as a valuation where the group is a subgroup of the real numbers isomorphic to the integers.

Domain/Measure theory definition

Let a topological space: a valuation is any map



satisfying the following three properties



The definition immediately shows the relationship between a valuation and a measure: the properties of the two mathematical object are often very similar if not identical, the only difference being that the domain of a measure is the Borel algebra of the given topological space, while the domain of a valuation is the class of open sets. Further details and references can be found in and .

Continuous valuation

A valuation (as defined in domain/measure theory) is said to be continuous if for every directed family of open sets (i.e. an indexed family of open sets which is also directed in the sense that for each pair of indexes and belonging to the index set , there exists an index such that and ) the following equality holds

Simple valuation

A valuation (as defined in domain/measure theory) is said to be simple if it is a finite linear combination with non-negative coefficients of Dirac valuations, i.e.



where is always greather than or al least equal to zero for all index . Simple valuations are obviously continuous in the above sense. The supremum of a directed family of simple valuations (i.e. an indexed family of simple valuations which is also directed in the sense that for each pair of indexes and belonging to the index set , there exists an index such that and ) is called quasi-simple valuation

Related topics

Examples

All the following examples, except the first and the last one, deal with Dedekind valuations: all shown valuations, except the last one, are surjective.

Logical equality

A very simple example of valuation, illustrating a basic part of the process of formalization of logical arguments using mathematical symbols, is the following: the statement

""


is satisfied by (i.e. true for) every valuations in which "" is mapped to the same value as "", and not satisfied by (i.e. false for) all other valuations.

-adic valuation

Let be a principal ideal domain, be its field of fractions, be one of its irreducible elements. Then, if the ideal is prime,



i.e. any element belongs to its -th power, for a proper natural number : this can be easily seen since
  • if , then belongs to for any natural number ,
  • if and share non trivial common factors, then belongs to , i.e. ,
  • if is coprime respect to , it is sufficient to choose : then


Therefore, any element of the field can be written as follows



where are coprime respect to , and is now an integer. Then the map defined as



is easily proven to be a valuation. When the principal ideal domain considered is the ring of integers, is a prime number , and this valuation is called -adic valuation on the set of rational numbers''.

-adic valuation

Let be a local integral ring with maximal ideal : then



i.e. every element of the local ring belongs to the -th power of its maximal ideal, for a proper natural number . Now define the map as



and extend it to the field of fractions of as follows:



It is easy to prove that this map is a well-defined valuation: it is called -adic valuation on . If, for example, the local integral ring considered is the ring of formal power series in two variables over the complex field i.e. , then its maximal ideal is and its -adic valuation is given by the difference of the orders of the power series in the numerator and the denominator: as examples, computation of -valuation for some fractions is reported

Geometric notion of contact

Let be the ring of polynomials of two variables over the complex field, be the field of rational functions over the same field, and consider the (convergent) power series



whose zero set, the analytic variety , can be parametrized by one coordinate as follows



It is possible to define a map as the value of the order of the formal power series in the variable obtained by restriction of any polynomial in to the points of the set ''



It is also possible to extend the map from its original ring of definition to the whole field as follows



As the power series is not a polynomial, it is easy to prove that the extended map is a valuation: the value is called intersection number between the curves (1-dimensional analytic varieties) and ''. As an example, the computation of some intersection numbers follows

Dirac valuation

Let a topological space, and let be a point of : the map
is a valuation in the domain/measure theory, sense called Dirac valuation. This concept bears its origin from distribution theory as it is an obvious transposition to valuation theory of Dirac distribution: as seen above, Dirac valuations are the "bricks" simple valuations are made of.

See also

Rererences

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Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration.
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This article discusses model theory as a mathematical discipline and not the informally used term mathematical model as used in other parts of mathematics and science.

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Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Arabic[1] mathematician, astronomer, astrologer and geographer,
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Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry.
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In mathematics, an algebraic number field (or simply number field) F is a finite, (and hence algebraic) field extension of the field of rational numbers Q.
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commutative algebra studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the
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In complex analysis, a pole of a holomorphic function is a certain type of singularity that behaves like the singularity 1/zn at z = 0.
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multiplicity of a member of a multiset is how many memberships in the multiset it has. For example, the term is used to refer to the number of times a given polynomial equation has a root at a given point.
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In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0.

Multiplicity of a zero

A complex number a is a simple zero of f, or a zero of multiplicity 1 of f
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In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. Equivalence classes are generally called jets.
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algebraic variety is essentially a (finite or infinite) set of points where a polynomial (in one or more variables) attains, or a set of such polynomials all attain, a value of zero.
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In mathematics, specifically geometry, an analytic variety is defined locally as the set of common solutions of several equations involving analytic functions. It is analogous to the included concept of complex algebraic variety, and any complex manifold is an analytic variety.
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Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry.
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In mathematics the concept of a measure generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base set, a "measure" is any consistent assignment of "sizes" to (some of) the subsets of the
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Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory.
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In mathematics and related technical fields, the term map or mapping is often a synonym for function. Thus, for example, a partial map is a partial function, and a total map is a total function.
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In topology and related fields of mathematics, a set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U.
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Topological spaces are mathematical structures that allow the formalization of concepts such as convergence, connectedness and continuity.
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A negative number is a number that is less than zero, such as −3. A positive number is a number that is greater than zero, such as 3. Zero itself is neither positive nor negative.
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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The word infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts (usually linked to the idea of "without end") which arise in philosophy, mathematics, and theology.
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In mathematics the concept of a measure generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base set, a "measure" is any consistent assignment of "sizes" to (some of) the subsets of the
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In mathematics the concept of a measure generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base set, a "measure" is any consistent assignment of "sizes" to (some of) the subsets of the
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Probability theory is the branch of mathematics concerned with analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities
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Theoretical computer science is the collection of topics of computer science that focuses on the more abstract, logical and mathematical aspects of computing, such as the theory of computation, analysis of algorithms and semantics of programming languages.
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