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.

Contents

Logic/Model theory definition

Algebraic definition

Equivalence of valuations

Dedekind valuation

Domain/Measure theory definition

Continuous valuation

Simple valuation

Related topics

Examples

Logical equality

''$p$''-adic valuation

''$mu$''-adic valuation

Geometric notion of contact

Dirac valuation

See also

Rererences

External links

Logic/Model theory definition

The starting point of the discourse is a given formal language
:
where '''$A$''' is its alphabet, '''$R$''' a set of transformation rules on '''$A$''', and '''$F$''' the closure of '''$A$''' under the elements of '''$R$''' — i.e. the set of (well-formed) formulas.
Given an abstract algebra $scriptstylemathfrak\{A\}$ 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 '$0$' or '$1$', i.e.
:
eg}
with properly defined ''logical disjunction'' , ''logical conjunction'' $scriptstylewedge$, ''logical implication'' $scriptstyleRightarrow$ and ''logical negation'' $scriptstyle$
eg, a 'valuation' is any map
:
ightarrowmathfrak{A} iff v in mathfrak{A}^oldsymbol{V_0}
where '''$V\_0$''' is the set of propositional variables of the language '''$L$''' . Thus, a valuation maps propositional variables to algebraic formulas in $scriptstylemathfrak\{A\}$: details on logic concepts can be found in .

Algebraic definition

To define the algebraic concept of valuation, the following objects are needed:

★ a field $scriptstylemathbb$ and its multiplicative subgroup $scriptstylemathbb^$
★

★ a commutative ordered group $scriptstyle(mathfrak,+,geq)$ which can be given in multiplicative notation as $scriptstyle(mathfrak,cdot,geq)$
and also an element $scriptstyleinfty$ such that
:$$
egin{array}{cr}
infty geq mathfrak & orallmathfrakinmathfrak{G}cup{infty} \
infty+mathfrak{a} = mathfrak{a} + infty = infty & orallmathfrak{a}inmathfrak{G}cup{infty}
end{array}

Then a 'valuation' is any map
:$v:mathbb\{K\}$
ightarrow {mathfrak{G} cup {infty}}
which satisfies the following properties
$$
egin{array}{ll}
v(a)=infty & mbox{iff}~a=0 \
v(ab)=v(a)+v(b) & orall a,binmathbb{K}^
★ \
v(a+b)geqmin{v(a),v(b)} & orall a,binmathbb{K}
end{array}

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 $scriptstyleinfty$, an element $scriptstylemathfrak\{0\}$ is given such that
:$$
egin{array}{cr}
mathfrak{0} leq mathfrak & orallmathfrakinmathfrak{G}cup{mathfrak{0}} \
mathfrak{0}mathfrak{a} = mathfrak{a}mathfrak{0} = mathfrak{0} & orallmathfrak{a}inmathfrak{G}cup{mathfrak{0}}
end{array}

then a 'valuation' is any map
:$v:mathbb\{K\}$
ightarrow mathfrak{G cup {0}}
satisfying the following properties (written using the multiplicative notation for group operation)
$$
egin{array}{ll}
v(a)=mathfrak{0} & mbox{iff}~a=0 \
v(ab)=v(a)v(b) & orall a,binmathbb^
★ \
v(a+b)leqmax{v(a),v(b)} & orall a,binmathbb{K}
end{array}

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, $scriptstylemathfrak\{0\}$ 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 $scriptstylemathfrak\{0\}$ of the group $scriptstylemathfrak\{G\}$, then for each valuation $scriptstyle\; v,$
:$v\; =\; v\; +\; mathfrak\{0\}\; iff\; v\; propto\; v$,

★ It is symmetric since, being $scriptstylemathfrak\{G\}$ a group, it contains the inverse element of each of its elements, so
:$v\_1\; propto\; v\_2\; Rightarrow\; v\_1\; =\; v\_2\; +\; mathfrak\{c\}\; iff\; v\_2\; =\; v\_1\; +\; mathfrak\{c\}^\{-1\}\; Rightarrow\; v\_2\; propto\; v\_1$

★ It is transitive since, given three valuation $scriptstyle\; v\_1,v\_2,v\_3,$ such that $scriptstyle\; v\_1,$ is equivalent to $scriptstyle\; v\_2,$ which is in turn equivalent to $scriptstyle\; v\_3,$, then
:$$
egin{array}{l}
v_1 propto v_2 \
v_2 propto v_3
end{array}
Rightarrow
egin{array}{l}
v_1 = v_2 + mathfrak{c} \
v_2 = v_3 + mathfrak{k}
end{array}
Rightarrow; v_1 = v_2 + mathfrak{c} = (v_3 + mathfrak{k}) + mathfrak{c} = v_3 + (mathfrak{k} + mathfrak{c}) ;Rightarrow; v_1 propto v_3

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 $scriptstylemathbb\{Q\}$: these are precisely equivalence classes of valuations for the p-adic completions of $scriptstylemathbb$.

Dedekind valuation

A 'Dedekind valuation' is a valuation for which the ordered abelian group $scriptstylemathfrak\{G\}$ in its codomain is the additive group of the integers, i.e.
:$(mathfrak\{G\},+,leq)\; =\; (mathbb\{Z\},+,leq)$
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 $scriptstylemathfrak\{G\}$ is a subgroup of the real numbers ''isomorphic'' to the integers.

Domain/Measure theory definition

Let $scriptstyle\; (X,mathcal\{T\})$ a topological space: a 'valuation' is any map
:$v:mathcal\{T\}$
ightarrow mathbb{R}^+cup{+infty}
satisfying the following three properties
$$
egin{array}{lcl}
v(U) = 0 & mbox{if}~U=emptyset & scriptstyle{ ext{Strictness property}}\
v(U)leq v(V) & mbox{if}~Usubseteq Vquad U,Vinmathcal{T} & scriptstyle{ ext{Monotonicity property}}\
v(Ucup V)+ v(Ucap V) = v(U)+v(V) & orall U,Vinmathcal{T} & scriptstyle{ ext{Modularity property}},
end{array}

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'' $scriptstyle\; \{U\_i\}\_\{iin\; I\}$ ''of open sets'' (i.e. an indexed family of open sets which is also directed in the sense that for each pair of indexes $i$ and $j$ belonging to the index set $I$, there exists an index $k$ such that $scriptstyle\; U\_isubseteq\; U\_k$ and $scriptstyle\; U\_jsubseteq\; U\_k$) the following equality holds
:$v(cup\_\{iin\; I\}U\_i)\; =\; sup\_\{iin\; I\}\; v(U\_i)$

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 $a\_i$ is always greather than or al least equal to zero for all index $i$. 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 $i$ and $j$ belonging to the index set $I$, there exists an index $k$ such that $scriptstyle\; v\_i(U)leq\; v\_k(U)!$ and $scriptstyle\; v\_j(U)subseteq\; v\_k(U)!$) 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
:"$x\; =\; y,$"
is ''satisfied by'' (i.e. true for) every valuations in which "$x$" is mapped to the same value as "$y$", and ''not satisfied by'' (i.e. false for) all other valuations.

''$p$''-adic valuation

Let $scriptstylemathfrak$ be a principal ideal domain, $scriptstylemathbb$ be its field of fractions, $scriptstylemathfrak\; in\; mathfrak$ be one of its irreducible elements. Then, if the ideal $scriptstyle(mathfrak\{p\})$ is prime,
:
i.e. any element belongs to its ''$k$''-th power, for a proper natural number $k$: this can be easily seen since

★ if $scriptstylemathfrak$, then $scriptstylemathfrak\{g\}$ belongs to $scriptstyle(mathfrak\{p\})^k$ for any natural number $k$,

★ if $scriptstylemathfrak\{g\}$ and $scriptstylemathfrak\{p\}$ share non trivial common factors, then $scriptstylemathfrak\{g\}$ belongs to $scriptstyle(\; mathfrak\{p\}\; )$, i.e. $k\; =\; 1$,

★ if $scriptstylemathfrak\{g\}$ is coprime respect to $scriptstylemathfrak\{p\}$, it is sufficient to choose $k\; =\; 0$: then
:$mathfrak\{p^0=R\}\; iff\; mathfrak\{g\}\; in\; mathfrak\{p^0\}$
Therefore, any element $s$ of the field $scriptstylemathbb\{K\}$ can be written as follows
:$s\; =\; mathfrak\{q\}/mathfrak\{r\}\; cdot\; mathfrak\{p\}^k\; quad\; mathfrak\{q,rin\; R\},,,\; kinmathbb\{Z\}$
where $scriptstylemathfrak\{q,r\}$ are coprime respect to $scriptstylemathfrak\{p\}$, and $k$ is now an integer. Then the map $scriptstyle\; v:mathbb\{K\}$
ightarrow mathbb{Z} defined as
:$$
v(s) =
egin{cases}
k & orall s in mathbb{K}^
★ \
infty & s=0 in mathbb{K}
end{cases}

is easily proven to be a valuation. When the principal ideal domain considered is the ring of integers, $scriptstylemathfrak\{p\}$ is a prime number ''$p$'', and this valuation is called '''$p$''-adic valuation' on the set $scriptstylemathbb\{Q\}$ of rational numbers''.

''$mu$''-adic valuation

Let $scriptstyle(mathfrak,mu)$ be a local integral ring with maximal ideal ''$mu$'': then
:
i.e. ''every element of the local ring belongs to the ''$k$''-th power of its maximal ideal'', for a proper natural number ''$k$''. Now define the map $scriptstyle\; v:mathfrak\{R\}$
ightarrowmathbb{Z} as
:$v(mathfrak\{f\})\; =\; k,Longleftrightarrow,mathfrakinmu^k\; mbox\{but\}~mathfrak$
otinmu^quad orall mathfrak in mathfrak
and extend it to the field of fractions $scriptstylemathbb$ of $scriptstylemathfrak$ as follows:
:$$
vmathfrak{(f/g)}=
egin{cases}
v(mathfrak) - v(mathfrak) & orall mathfrak{f/g} in mathbb{K}^
★ \
infty & mathfrak{f}=mathfrak{0} in mathbb{K}
end{cases}

It is easy to prove that this map is a well-defined valuation: it is called '''$mu$''-adic valuation' on $scriptstylemathbb\{K\}$. If, for example, the local integral ring considered is the ring of formal power series in two variables over the complex field i.e. $scriptstylemathfrak\{R\}\; =\; mathbb\{C\}x,y$, then its maximal ideal is $scriptstylemu\; =\; (x;y)\; ,$
and its ''$mu$''-adic valuation is given by the difference of the orders of the power series in the numerator and the denominator: as examples, computation of ''$mu$''-valuation for some fractions is reported
:$v(x^2\; +\; y^2\; +\; x^3y^2)=2\; ,$
:$v(x^3/y^2)\; =\; 3\; -\; 2\; =\; 1\; ,$

Geometric notion of contact

Let $scriptstylemathfrak\; =\; mathbb[x,y]$ be the ring of polynomials of two variables over the complex field, $scriptstylemathbb\; =\; mathbb(x,y)$ be the field of rational functions over the same field, and consider the (convergent) power series
:
whose zero set, the analytic variety ''$scriptstyle\; V\_f,$'', can be parametrized by one coordinate ''$t$'' as follows
:$V\_f\; =\; \{(x,y)inmathbb\{C\}^2,|,\; f(x,y)\; =\; 0\}\; =\; left\{\; (x,y)inmathbb\{C\}^2,|,(x,y)\; =\; left(t,sum\_\{n=3\}^\{infty\}t^i$
ight)
ight}
It is possible to define a map $scriptstyle\; v:\; mathbb[x,y]$
ightarrow mathbb{Z} as ''the value of the order of the formal power series in the variable ''$t$'' obtained by restriction of any polynomial ''$P$'' in $scriptstylemathbb\{C\}[x,y]$ to the points of the set ''$scriptstyle\; V\_f,$''
:$$
v(P) = mathrm{ord}_tleft(P|_{V_f}
ight) = {mathrm{ord}}_t left(Pleft(t,sum_{n=3}^{+infty}t^i
ight)
ight) quad orall Pin mathbb{C}[x,y]

It is also possible to extend the map $v$ from its original ring of definition to the whole field $scriptstylemathbb(x,y)$ as follows
:$$
v(P/Q) =
egin{cases}
v(P) - v(Q) & orall P/Q in {mathbb{C}(x,y)}^
★ \
infty & P equiv 0 in mathbb{C}(x,y)
end{cases}

As the power series ''$f$'' is not a polynomial, it is easy to prove that the extended map ''$v$'' is a valuation: the value ''$scriptstyle\; v(P)$ is called ''intersection number between the curves (1-dimensional analytic varieties) ''$scriptstyle\; V\_P,$'' and $scriptstyle\; V\_f,$''. As an example, the computation of some intersection numbers follows
:$$
egin{array}{l}
v(x) = mathrm{ord}_t(t) = 1 \
v(x^6-y^2)=mathrm{ord}_t(t^6-t^6-2t^7-3t^8-dots)=mathrm{ord}_t (-2t^7-3t^8-dots)=7 \
vleft(rac{x^6 - y^2}{x}
ight)= mathrm{ord}_t (-2t^7-3t^8-dots) - mathrm{ord}_t(t) = 7 - 1 = 6
end{array}

Dirac valuation

Let $scriptstyle\; (X,mathcal\{T\})$ a topological space, and let ''$x$'' be a point of ''$X$'': the map
:$delta\_x(U)=$
egin{cases}
0 & mbox{if}~x
otin U\
1 & mbox{if}~xin U
end{cases}
quadorall Uinmathcal{T}

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

★ Absolute value (algebra).

★ Discrete valuation.

★ Domain theory.

★ Local ring.

★ Measure (mathematics).

★ Ostrowski's theorem.

★ Valuation ring.

Rererences

★ . The preprint from the homepage of the second author is freely readable.

★ . Published as "''Extension of valuations''", Mathematical Structures in Computer Science (2005), 15: 271-297, DOI:10.1017/S096012950400461X.

★ , ISBN 0-7167-1933-9, chapter 9 paragraph 6 ''Valuations''.

★ , 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.