Subsystem

Definition

Let $\mathcal{S}=(S,{\Gamma }_{\mathfrak{P}},\mathbf{R},\mathbf{V})$ be a system. Consider the set: $\mathfrak{R}=\left({\bigcup }_k\mathbf{R}\right)\cup \left({\bigcup }_i{\bigcup }_k\mathbf{V}\right)$

A subsystem ${\mathcal{S}}^{\prime }=(S^{\prime },{\Gamma }_{{\mathfrak{P}}^{\prime }}^{\prime },{\mathbf{R}}^{\prime },{\mathbf{V}}^{\prime })$ of $\mathcal{S}$ is a system such that:

(i) $S^{\prime }\subseteq S$ and $\bigcup {\Gamma }_{{\mathfrak{P}}^{\prime }}^{\prime }\subseteq \bigcup {\Gamma }_{\mathfrak{P}}$

(ii) there exist families of functions $\{f_n\},\{g_m\}$ with $n,m\le |{\Gamma }_{{\mathfrak{P}}^{\prime }}^{\prime }|=C^{\prime }$, such that, for some $p_n,q_n$, $f_n:{\mathfrak{R}}^{p_n}\to {\Gamma }_{{\mathfrak{P}}^{\prime }}^{\prime q_n}$ and for $p_m,q_m$, $g_m:{\mathfrak{R}}^{p_m}\to \mathcal{P}({\Gamma }_{{\mathfrak{P}}^{\prime }}^{\prime q_m}\times V_{kij}^{\prime })$, for given sets of values $V_{kij}^{\prime }$, satisfying:

• (a) for any element $R^{\prime }$ of ${\bigcup }_k{\mathbf{R}}^{\prime }$ there is exactly one $n$ such that for some $\mathbf{\sigma }\in {\mathfrak{R}}^{p_n}$, $f_n(\mathbf{\sigma })=R^{\prime }$

• (b) for any element ${\mathrm{v}}^{\prime }$ of ${\bigcup }_k{\bigcup }_i{\mathbf{V}}^{\prime }$, if $\mathrm{dom}({\mathrm{v}}^{\prime })=R^{\prime }$ for some $R^{\prime }\in {\bigcup }_k{\mathbf{R}}^{\prime }$, then there is exactly one $m$ such that for some $\mathbf{\sigma }\in {\mathfrak{R}}^{p_m}$, $g_m(\mathbf{\sigma })={\mathrm{v}}^{\prime }$

We denote the system-inclusion relation by $⊏$.

Interpretation

This definition allows as few restrictions as possible on the relationship between systems and subsystems. A subsystem has an underlying set bounded by the underlying set of its supersystem, and its elements’ parts on the underlying set are also bounded by the parts of the elements of the supersystem.

The relations and valuations of subsystems can depend on arbitrary relations and valuations over the corresponding supersystem. This enables describing emergent properties: relations of any arity on a subsystem can sensibly depend on relations of any other arity on its supersystem.

Key Characteristics

• Forms a coherent system in its own right
• Part of a larger encompassing system
• Maintains internal structure and relations
• Can be analyzed independently
• Interacts with other subsystems
• May be recursively composed of sub-subsystems
• Enables hierarchical system organization
• Facilitates modular analysis and design

Types of Subsystems

1. Functional Subsystems:

   • Defined by specific function or purpose
   • Example: Circulatory subsystem in organism

2. Structural Subsystems:

   • Defined by physical or structural boundaries
   • Example: Engine subsystem in vehicle

3. Behavioral Subsystems:

   • Defined by patterns of behavior
   • Example: Decision-making subsystem in organization

Subsystem Relations

Subsystems can relate to each other through:

• Interface: Shared boundaries and interaction points
• Coupling: Degree of interdependence
• Coordination: Synchronized behavior
• Hierarchy: Nested containment relationships

Examples

1. Biological Organism:

   • System: Human body
   • Subsystems: Nervous system, circulatory system, respiratory system
   • Sub-subsystems: Heart, lungs, brain

2. Computer System:

   • System: Desktop computer
   • Subsystems: Processing unit, storage subsystem, I/O subsystem
   • Sub-subsystems: CPU cache, RAM modules, disk drives

3. Organization:

   • System: Corporation
   • Subsystems: Sales department, engineering department, finance department
   • Sub-subsystems: Product teams, regional offices

4. Ecosystem:

   • System: Forest ecosystem
   • Subsystems: Canopy layer, understory, soil system
   • Sub-subsystems: Specific microhabitats, microbial communities

Formal Properties

For subsystems S₁ and S₂ of system S:

1. Closure: A subsystem of a subsystem is a subsystem of the original
2. Union: The union of subsystems may form a larger subsystem
3. Intersection: The intersection of subsystems is a subsystem
4. Minimal: Single elements can be considered minimal subsystems
5. Maximal: The entire system is the maximal subsystem of itself

Design Principles

Effective subsystem decomposition should:

• Maximize cohesion within subsystems
• Minimize coupling between subsystems
• Define clear interfaces and boundaries
• Enable independent analysis and modification
• Support hierarchical understanding
• Facilitate parallel development

Key References

Theory of Hierarchical, Multilevel, Systems

Mihajlo D. Mesarović, D. Macko, Yasuhiko Takahara (1970) View in Zotero Library

Discusses subsystems extensively in the context of hierarchical and multilevel systems, establishing the theoretical foundation for subsystem decomposition.

General Systems Theory: Mathematical Foundations

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

Provides the mathematical formalization of subsystem relations and their role in system decomposition and hierarchical organization.

Related Concepts

• system - The whole of which a subsystem is part
• nested-system - Systems organized in containment hierarchies
• hierarchical-decomposition - Process of breaking systems into subsystems
• hierarchy - Organizational structure of systems and subsystems
• input-output-system - Subsystems as I/O transformations
• relational-structure - Mathematical foundation

Bibliography Keys

• mesarovic1970theory
• mesarovic1975general
• wymore1967systems
• klir1985architecture
• simon1962architecture