System

Definition

Let $S$ be a set with cardinality $|S|\ge 2$. Consider a partition $\mathfrak{P}$ of $S$, and a subset ${\Gamma }_{\mathfrak{P}}$ of $\mathfrak{P}$, with $|{\Gamma }_{\mathfrak{P}}|=C$.

Define $\mathbf{R}=\{R_k|k\in {\mathbb{N}}_{\le C}\}$ where each $R_k$ is a set of $n_k+1$ distinct $k$-ary relations over ${\Gamma }_{\mathfrak{P}}$, i.e., for any $R_{ki}\in R_k$ we have $R_{ki}\subseteq {\Gamma }_{\mathfrak{P}}^k$ and therefore: $R_k=\{R_{ki}\subseteq {\Gamma }_{\mathfrak{P}}^k|i\in {\mathbb{N}}_{\le n_k}\}$

Define, now, for each $R_k$ and each relation $R_{ki}\in R_k$, a set $P_{ki}$ (possibly empty) of functions ${\mathrm{v}}_{kij}$ such that, for each $j$, ${\mathrm{v}}_{kij}:R_{ki}\to V_{kij}$, where $V_{kij}$ is the set of values for the valued relation ${\mathrm{v}}_{kij}(R_{ki})$.

Finally, define the set $\mathbf{V}=\{P_{ki}|k\in {\mathbb{N}}_{\le C};i\in {\mathbb{N}}_{\le n_k}\}$ of sets of valuations. Then a system $\mathcal{S}$ is a quadruple: $\mathcal{S}=(S,{\Gamma }_{\mathfrak{P}},\mathbf{R},\mathbf{V})$

We call the set $S$ the underlying set of $\mathcal{S}$, denoted by $\mathfrak{u}(\mathcal{S})$. We call ${\Gamma }_{\mathfrak{P}}$ the set of elements of $\mathcal{S}$, denoted by $\mathfrak{e}(\mathcal{S})$. We call $\mathbf{R}$ the set of relations of $\mathcal{S}$, denoted $\mathfrak{r}(\mathcal{S})$, and $\mathbf{V}$ the set of valuations of $\mathcal{S}$, denoted $\mathfrak{v}(\mathcal{S})$.

We say that a non-empty set $s\subset S$ is an element of a system $\mathcal{S}$ iff $s\in {\Gamma }_{\mathfrak{P}}$, and denote the system-membership relation by $\text{sqin}$. Thus: $s\text{sqin}\mathcal{S}⟺s\in {\Gamma }_{\mathfrak{P}}$

Alternative Definitions

Mesarovic and Takahara’s Definition: A system $S$ is a relation on nonempty sets: $S\subseteq \prod \{V_i:i\in I\}$

Yi Lin’s Multirelation Approach: A system is an ordered pair $S=(M,R)$ where $M$ is a set of objects and $R$ is a set of relations on $M$.

Key Characteristics

• Composed of interrelated parts or components
• Forms a cohesive, identifiable whole
• Exhibits properties that emerge from component interactions
• Can be analyzed at multiple levels of abstraction
• Has boundaries that distinguish it from its environment
• May exchange matter, energy, or information with environment
• Can be decomposed into subsystems
• Behavior determined by structure and relationships

Types of Systems

Systems can be classified along several dimensions:

1. Openness:

   • Open systems: Exchange with environment
   • Closed systems: No exchange with environment
   • Isolated systems: No interaction whatsoever

2. Complexity:

   • Simple systems: Few components, clear relationships
   • Complicated systems: Many components, deterministic
   • Complex systems: Many components, emergent behavior

3. Nature:

   • Physical systems: Material components
   • Abstract systems: Conceptual or mathematical
   • Social systems: Human interactions
   • Biological systems: Living organisms

Examples

1. Solar System:

   • Objects: Sun, planets, moons, asteroids
   • Relations: Gravitational attraction, orbital relationships
   • Properties: Mass, position, velocity

2. Ecosystem:

   • Objects: Species, organisms, resources
   • Relations: Predator-prey, competition, symbiosis
   • Properties: Population, energy flow, nutrient cycles

3. Computer System:

   • Objects: CPU, memory, storage, I/O devices
   • Relations: Data flow, control signals, dependencies
   • Properties: Performance, capacity, state

4. Organization:

   • Objects: Departments, employees, resources
   • Relations: Reporting structure, communication, workflow
   • Properties: Roles, responsibilities, capabilities

Formal Representation

A system S can be formally represented as:

• S = (O, R) where O is a set of objects and R is a set of relations on O
• Or as a relational structure: S = ⟨D, R₁, R₂, …, Rₙ⟩
• Or in Mesarovic’s framework as a family of valued relations

Key References

General Systems Theory: Mathematical Foundations

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

Provides the canonical set-theoretic definition of systems as relations on Cartesian products: $S\subseteq \prod \{V_i:i\in I\}$.

General Systems Theory: A Mathematical Approach

Yi Lin (1999) View in Zotero Library

Introduces the multirelation approach $S=(M,R)$ emphasizing that “the concept of systems is a generalization of that of structures.”

CONCEPTUAL BASIS FOR A MATHEMATICAL THEORY OF GENERAL SYSTEMS

Mihajlo D. Mesarovic (1972) View in Zotero Library DOI: 10.1108/eb005295

Reappraises foundations of systems theory, sharpening basic systems concepts and introducing goal-seeking representations within set-theory and abstract mathematics.

Related Concepts

• subsystem - Components within a system
• nested-system - Systems containing other systems
• relational-structure - Mathematical foundation
• valued-relation - Formal representation of system relations
• input-output-system - Systems viewed through I/O behavior
• hierarchy - Organizational structure of systems

Bibliography Keys

• bertalanffy1968general
• mesarovic1975general
• klir1985architecture
• wymore1967systems