# Valuation (mathematics)

Informally, a

In logic and model theory, a

In algebra (in particular in algebraic geometry or algebraic number theory), a

In measure theory or at least in the approach to it though domain theory, a

with properly defined

where

Then a

which satisfies the following properties

Note that some authors use the term

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

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

then a

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,

Every equivalence class of valuations over a field with respect to this equivalence relation is called a

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 .

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

is

i.e. any element belongs to its -th power, for a proper natural number : this can be easily seen since

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

i.e.

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

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

It is possible to define a map as

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

An

**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**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 mapwhere

**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:- a field and its multiplicative subgroup
- a commutative ordered group which can be given in multiplicative notation as

Then a

**valuation**is any mapwhich 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 mapsatisfying 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 reflexive since, considering the neutral element of the group , then for each valuation

- ,

- It is symmetric since, being a group, it contains the inverse element of each of its elements, so

- 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.**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 mapsatisfying 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

- The
**extension problem**for a given valuation (in the sense of domain/measure theory) consists in finding under what type of conditions it can be extended to a measure on a proper topological space, which may or may not be the same space where it is defined: the papers and in the reference section are devoted to this aim and give also several historical details. - The concepts of
**valuation on convex sets**and**valuation on manifolds**are a generalization of valuation in the sense of domain/measure theory. A valuation on convex sets is allowed to assume complex values, and the underlying topological space is the set of non-empty convex compact subsets of a finite-dimensional vector space: a valuation on manifolds is a complex valued finitely additive measure defined on a proper subset of the class of all compact submanifolds of the given manifolds. Details can be found in several arxiv papers of prof. Seymon Alesker.

## 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 : theni.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 asand 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 serieswhose 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**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

- Absolute value (algebra).
- Discrete valuation.
- Domain theory.
- Local ring.
- Measure (mathematics).
- Ostrowski's theorem.
- Valuation ring.

## Rererences

- Alvarez-Manilla, Maurizio; Abbas Edalat & Nasser Saheb-Djahromi (2000), "An extension result for continuous valuations",
*Journal of the London Mathematical Society*61: 629-640, DOI:10.1112/S0024610700008681. The preprint from the homepage of the second author is freely readable. - Goulbault-Larrecq, Jean (November 2002),
*Extensions of valuations*, France: LSV (Laboratoire de Spécification at Vérification),CNRS & ENS de Cachan. Published as "*Extension of valuations*", Mathematical Structures in Computer Science (2005), 15: 271-297, DOI:10.1017/S096012950400461X. - Jacobson, Nathan (1989),
*Basic algebra II*(2^{nd}ed.), New York: W. H. Freeman, ISBN 0-7167-1933-9, chapter 9 paragraph 6*Valuations*. - Rasiowa, Helena & Roman Sikorski (1970),
*The Mathematics of Metamathematics*(3^{rd}ed.), Warsaw: PWN, chapter 6*Algebra of formalized languages*.

## External links

- "
*Discrete valuation*", Planetmath.org Encyclopedia. - "
*Valuation*", Planetmath.org Encyclopedia. - Alesker, Seymon, "
*various preprints on valuations*", arxiv preprint server, primary site at Cornell University. Several papers dealing with valuations on convex sets, valuations on manifolds and related topics. - V.I. Danilov "
*Valuation*" Springer-Verlag Online Encyclopaedia of Mathematics. - Weisstein, Eric W., "
*Valuation*from MathWorld--A Wolfram Web Resource.

**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**.**.....**Click the link for more information.**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/*z*^{n}at*z*= 0.**.....**Click the link for more information.**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***.....**Click the link for more information. 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.**.....**Click the link for more information.**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.**.....**Click the link for more information.**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**.....**Click the link for more information.**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*.**.....**Click the link for more information. 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*.**.....**Click the link for more information.For a general, non-technical overview of the subject, see .

**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.**.....**Click the link for more information. 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**.....**Click the link for more information. 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.**.....**Click the link for more information. 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**.....**Click the link for more information. 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**.....**Click the link for more information.**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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*This article is about the term***formal language**as it is used in mathematics, logic and computer science. For information about a mode of expression that is more disciplined or precise than everyday speech, see Register (linguistics).

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*ABCs redirects here, for the***A**lien**B**ig**C**at**s**, see British big cats.

An

**alphabet**is a standardized set of

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**SET**may stand for:

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*This article is about the term***formal language**as it is used in mathematics, logic and computer science. For information about a mode of expression that is more disciplined or precise than everyday speech, see Register (linguistics).

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