Valuation (mathematics)

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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.

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[edit] Logic/Model theory definition

The starting point of the discourse is a given formal language

\boldsymbol{L}=\{\boldsymbol{A},\boldsymbol{R},\boldsymbol{F}\}

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 \scriptstyle\mathfrak{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.

\mathfrak{A}\equiv\{\boldsymbol{F},\vee,\wedge,\Rightarrow,\neg\}

with properly defined logical disjunction \scriptstyle\vee, logical conjunction \scriptstyle\wedge, logical implication \scriptstyle\Rightarrow and logical negation \scriptstyle\neg, a valuation is any map

v:\boldsymbol{V_0}\rightarrow\mathfrak{A} \iff v \in \mathfrak{A}^\boldsymbol{V_0}

where V0 is the set of propositional variables of the language L . Thus, a valuation maps propositional variables to algebraic formulas in \scriptstyle\mathfrak{A}: details on logic concepts can be found in Rasiowa & Sikorski 1970.

[edit] Algebraic definition

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

and also an element \scriptstyle\infty such that

\begin{array}{cr} \infty \geq \mathfrak{{a}} & \forall\mathfrak{{a}}\in\mathfrak{G}\cup\{\infty\} \\ \infty+\mathfrak{a} = \mathfrak{a} + \infty = \infty & \forall\mathfrak{a}\in\mathfrak{G}\cup\{\infty\} \end{array}

Then a valuation is any map

v:\mathbb{K} \rightarrow {\mathfrak{G} \cup \{\infty\}}

which satisfies the following properties

\begin{array}{ll} v(a)=\infty & \mbox{iff}~a=0 \\ v(ab)=v(a)+v(b) & \forall a,b\in\mathbb{K}^* \\ v(a+b)\geq\min\{v(a),v(b)\} & \forall a,b\in\mathbb{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 \scriptstyle\infty, an element \scriptstyle\mathfrak{0} is given such that

\begin{array}{cr} \mathfrak{0} \leq \mathfrak{{a}} & \forall\mathfrak{{a}}\in\mathfrak{G}\cup\{\mathfrak{0}\} \\ \mathfrak{0}\mathfrak{a} = \mathfrak{a}\mathfrak{0} = \mathfrak{0} & \forall\mathfrak{a}\in\mathfrak{G}\cup\{\mathfrak{0}\} \end{array}

then a valuation is any map

v:\mathbb{K} \rightarrow \mathfrak{G \cup \{0\}}

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

\begin{array}{ll} v(a)=\mathfrak{0} & \mbox{iff}~a=0 \\ v(ab)=v(a)v(b) & \forall a,b\in\mathbb{{K}}^* \\ v(a+b)\leq\max\{v(a),v(b)\} & \forall a,b\in\mathbb{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, \scriptstyle\mathfrak{0} is the identity element the ordered group, or the zero element of the ring considered. See Jacobson 1989 for further details.

[edit] 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

v_1 \propto v_2 \iff \exists\mathfrak{c}\in\mathfrak{G}\;|\; v_1(x) = v_2(x) + \mathfrak{c}\quad\forall x\in\mathbb{K}

Proportionality in this sense is an equivalence relation:

v = v + \mathfrak{0} \iff v \propto v,
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
\begin{array}{l} v_1 \propto v_2 \\ v_2 \propto v_3 \end{array} \Rightarrow \begin{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 \scriptstyle\mathbb{Q}: these are precisely equivalence classes of valuations for the real and p-adic completions of \scriptstyle\mathbb{{Q}}.

[edit] Dedekind valuation

A Dedekind valuation is a valuation for which the ordered abelian group \scriptstyle\mathfrak{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 \scriptstyle\mathfrak{G} is a subgroup of the real numbers isomorphic to the integers.

[edit] Domain/Measure theory definition

Let \scriptstyle (X,\mathcal{T}) a topological space: a valuation is any map

v:\mathcal{T} \rightarrow \mathbb{R}^+\cup\{+\infty\}

satisfying the following three properties

\begin{array}{lcl} v(U) = 0 & \mbox{if}~U=\emptyset & \scriptstyle{\text{Strictness property}}\\ v(U)\leq v(V) & \mbox{if}~U\subseteq V\quad U,V\in\mathcal{T} & \scriptstyle{\text{Monotonicity property}}\\ v(U\cup V)+ v(U\cap V) = v(U)+v(V) & \forall U,V\in\mathcal{T} & \scriptstyle{\text{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 Alvarez-Manilla et al. 2000 and Goulbault-Larrecq 2002.

[edit] Continuous valuation

A valuation (as defined in domain/measure theory) is said to be continuous if for every directed family \scriptstyle \{U_i\}_{i\in I} of open sets (i.e. a 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_i\subseteq U_k and \scriptstyle U_j\subseteq U_k) the following equality holds

v(\cup_{i\in I}U_i) = \sup_{i\in I} v(U_i)

[edit] 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.

v(U)=\sum_{i=1}^n a_i\delta_{x_i}(U)\quad\forall U\in\mathcal{T}

where ai 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. a 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

\bar{v}(U) = \sup_{i\in I}v_i(U) \quad \forall U\in \mathcal{T}

[edit] Related topics

[edit] 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.

[edit] 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.

[edit] p-adic valuation

Let \scriptstyle\mathfrak{{R}} be a principal ideal domain, \scriptstyle\mathbb{{K}} be its field of fractions, \scriptstyle\mathfrak{{p}} \in \mathfrak{{R}} be one of its irreducible elements. Then, if the ideal \scriptstyle(\mathfrak{p}) is prime,

\mathfrak{g} \in (\mathfrak{{p}})^k \quad \forall \mathfrak{{g\in R}}, k\in\mathbb{N}

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

  • if \scriptstyle\mathfrak{{g = 0}}, then \scriptstyle\mathfrak{g} belongs to \scriptstyle(\mathfrak{p})^k for any natural number k,
  • if \scriptstyle\mathfrak{g} and \scriptstyle\mathfrak{p} share non trivial common factors, then \scriptstyle\mathfrak{g} belongs to \scriptstyle( \mathfrak{p} ), i.e. k = 1,
  • if \scriptstyle\mathfrak{g} is coprime respect to \scriptstyle\mathfrak{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 \scriptstyle\mathbb{K} can be written as follows

s = \mathfrak{q}/\mathfrak{r} \cdot \mathfrak{p}^k \quad \mathfrak{q,r\in R}\,,\, k\in\mathbb{Z}

where \scriptstyle\mathfrak{q,r} are coprime respect to \scriptstyle\mathfrak{p}, and k is now an integer. Then the map \scriptstyle v:\mathbb{K} \rightarrow \mathbb{Z} defined as

v(s) = \begin{cases} k & \forall 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, \scriptstyle\mathfrak{p} is a prime number p, and this valuation is called p-adic valuation on the set \scriptstyle\mathbb{Q} of rational numbers.

[edit] μ-adic valuation

Let \scriptstyle(\mathfrak{{R}},\mu) be a local integral ring with maximal ideal μ: then

\mathfrak{f} \in (\mu)^k \quad \forall \mathfrak{f}\in\mathfrak{{R}}\,,\,k\in\mathbb{N}

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}\rightarrow\mathbb{Z} as

v(\mathfrak{f}) = k\,\Longleftrightarrow\,\mathfrak{{f}}\in\mu^k\ \mbox{but}~\mathfrak{{f}}\notin\mu^{{k+ 1}}\quad \forall \mathfrak{{f}} \in \mathfrak{{R}}

and extend it to the field of fractions \scriptstyle\mathbb{{K}} of \scriptstyle\mathfrak{{R}} as follows:

v\mathfrak{(f/g)}= \begin{cases} v(\mathfrak{{f}}) - v(\mathfrak{{g}}) & \forall \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 μ-adic valuation on \scriptstyle\mathbb{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. \scriptstyle\mathfrak{R} = \mathbb{C}[[x,y]], then its maximal ideal is \scriptstyle\mu = (x;y) \, and its μ-adic valuation is given by the difference of the orders of the power series in the numerator and the denominator: as examples, computation of μ-valuation for some fractions is reported

v(x^2 + y^2 + x^3y^2)=2 \,
v(x^3/y^2) = 3 - 2 = 1 \,

[edit] Geometric notion of contact

Let \scriptstyle\mathfrak{{R}} = \mathbb{{C}}[x,y] be the ring of polynomials of two variables over the complex field, \scriptstyle\mathbb{{K}} = \mathbb{{C}}(x,y) be the field of rational functions over the same field, and consider the (convergent) power series

f(x,y) = y - \sum_{n=3}^{\infty} \frac{x^n}{n!} \in \mathbb{{C}}\{x,y\}

whose zero set, the analytic variety \scriptstyle V_f\,, can be parametrized by one coordinate t as follows

V_f = \{(x,y)\in\mathbb{C}^2\,|\, f(x,y) = 0\} = \left\{ (x,y)\in\mathbb{C}^2\,|\,(x,y) = \left(t,\sum_{n=3}^{\infty}t^i\right)\right\}

It is possible to define a map \scriptstyle v: \mathbb{{C}}[x,y] \rightarrow \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 \scriptstyle\mathbb{C}[x,y] to the points of the set \scriptstyle V_f\,

v(P) = \mathrm{ord}_t\left(P|_{V_f}\right) = {\mathrm{ord}}_t \left(P\left(t,\sum_{n=3}^{+\infty}t^i\right)\right) \quad \forall P\in \mathbb{C}[x,y]

It is also possible to extend the map v from its original ring of definition to the whole field \scriptstyle\mathbb{{C}}(x,y) as follows

v(P/Q) = \begin{cases} v(P) - v(Q) & \forall 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

\begin{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 \\ v\left(\frac{x^6 - y^2}{x}\right)= \mathrm{ord}_t (-2t^7-3t^8-\dots) - \mathrm{ord}_t(t) = 7 - 1 = 6 \end{array}

[edit] Dirac valuation

Let \scriptstyle (X,\mathcal{T}) a topological space, and let x be a point of X: the map

\delta_x(U)= \begin{cases} 0 & \mbox{if}~x\notin U\\ 1 & \mbox{if}~x\in U \end{cases} \quad\forall U\in\mathcal{T}

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

[edit] See also

[edit] Rererences

[edit] External links

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