Valued Relation

Definition

Let $\mathcal{F}=\{S_i{\}}_{i\in I}$ be a family of sets and $R$ an $n$-ary relation over $\mathcal{F}$, that is, $R\subseteq {\prod }_{i\in I}{S}_i$. Let $V$ be another set, called the set of values. If $\mathrm{v}$ is a function such that $\mathrm{v}:R\to V$, we define:

(a) $\mathrm{v}(R)=\{\mathrm{v}(r)\in V|r\in R\}$

(b) $R_{\mathrm{v}}=(R,V,\mathrm{v})$ is called an $n$-ary $\mathrm{v}$-valued relation over $\mathcal{F}$

(c) $\mathrm{v}$ is a valuation of $R$

If $R_{\mathrm{v}}=(R,V,\mathrm{v})$, we say that $R$ is the underlying relation of $R_{\mathrm{v}}$.

Interpretation

In the above definition, the graph of $\mathrm{v}$ is considered as the valued relation. Thus an $n$-ary valued relation over a family $\{S_i\}$ is an $(n+1)$-ary relation with the particularity that it is functional in $V$. The underlying relations account for the ‘structure of interactions between objects’, whereas the valuations denote the characteristics these interactions have, notably their strength or intensity.

As an example, a weighted digraph is an instance of a ‘structure of interactions with values’.

Key Characteristics

• Generalizes binary (yes/no) relations to multi-valued relations
• Assigns elements from a value set to tuples
• Value set can be any set (real numbers, probabilities, fuzzy values, etc.)
• Provides mathematical foundation for Mesarovic’s systems theory
• Enables quantitative system descriptions
• Supports probabilistic and fuzzy system modeling
• More expressive than classical relations

Special Cases

1. Classical Relation: Value set Y = {0, 1} or {true, false}
2. Fuzzy Relation: Value set Y = [0, 1] (degrees of membership)
3. Probabilistic Relation: Value set Y = [0, 1] (probabilities)
4. Weighted Relation: Value set Y = ℝ (real-valued weights)
5. Labeled Relation: Value set Y = any finite set of labels

Examples

1. Distance Relation:

   • Sets: Cities × Cities
   • Value set: ℝ⁺ (non-negative reals)
   • ρ(CityA, CityB) = distance between cities in km

2. Similarity Relation:

   • Sets: Objects × Objects
   • Value set: [0, 1]
   • ρ(obj₁, obj₂) = degree of similarity

3. Preference Relation:

   • Sets: Alternatives × Alternatives
   • Value set: {strongly prefer, prefer, indifferent, …}
   • ρ(a, b) = preference of a over b

4. Interaction Strength:

   • Sets: Neurons × Neurons
   • Value set: ℝ (positive or negative)
   • ρ(n₁, n₂) = synaptic weight from n₁ to n₂

5. Traffic Flow:

   • Sets: Locations × Locations × Time
   • Value set: ℕ (number of vehicles)
   • ρ(loc₁, loc₂, t) = vehicles traveling from loc₁ to loc₂ at time t

Formal Properties

Valued relations can exhibit various properties:

1. Reflexivity: ρ(x, x) has specific value for all x
2. Symmetry: ρ(x, y) = ρ(y, x)
3. Transitivity: Composite relation satisfies transitivity constraint
4. Completeness: Defined for all tuples in domain

Role in Systems Theory

In Mesarovic’s General Systems Theory, systems are defined as families of valued relations:

S = {ρᵢ | i ∈ I}

where each ρᵢ is a valued relation on some family of sets. This provides:

• Unified framework for diverse system types
• Mathematical rigor
• Ability to model uncertainty and gradations
• Foundation for hierarchical systems theory

Operations on Valued Relations

1. Composition: Combining relations through intermediate sets
2. Projection: Restricting to subset of argument positions
3. Join: Combining relations on common domains
4. Aggregation: Combining multiple value assessments

Key References

General Systems Theory: Mathematical Foundations

Mihajlo D. Mesarović, Yasuhiko Takahara (1975) View in Zotero Library

Central to Mesarovic and Takahara’s formalization, providing the mathematical foundation for valued relations as the basis of general systems theory.

Related Concepts

• relational-structure - Classical relations as special case
• system - Systems defined via valued relations
• set-theoretic-system - Set-theoretic foundations
• input-output-system - I/O behavior as valued relations
• fuzzy-set - Related concept using [0,1] valuations

Bibliography Keys

• mesarovic1975general
• mesarovic1989abstract
• klir1985architecture
• zadeh1965fuzzy (for fuzzy relations)