Supersystem

Definition

Let $\mathfrak{S}=\{{\mathcal{S}}_i{\}}_{i\in I}$ be a family of systems. Define, for each ${\mathcal{S}}_i$, the set:

${\mathfrak{R}}_i=\left(\bigcup \mathfrak{r}({\mathcal{S}}_i)\right)\cup \left(\bigcup \bigcup \mathfrak{v}({\mathcal{S}}_i)\right)$

Define $\mathfrak{R}={\bigcup }_{i\in I}\{{\mathfrak{R}}_i\}$. A system $\mathcal{S}$ is said a supersystem of each ${\mathcal{S}}_i$ (and, by extension, of $\mathfrak{S}$) if it satisfies:

i) $\bigcup _{s\in \mathfrak{S}}\mathfrak{u}(s)\subseteq \mathfrak{u}(\mathcal{S})$ (underlying sets aggregate)

ii) $\bigcup _{s\in \mathfrak{S}}\mathfrak{e}(s)\subseteq \mathfrak{e}(\mathcal{S})$ (elements aggregate)

iii) For each $R\in \bigcup \mathfrak{r}(\mathcal{S})$, there is a natural number $p\le |\mathfrak{R}|$ and a function $f$ such that, for some $\mathbf{\sigma }\in {\mathfrak{R}}^p$, $R=f(\mathbf{\sigma })$ (relations depend functionally on subsystem relations)

iv) For each $V\in \bigcup \bigcup \mathfrak{v}(\mathcal{S})$, there is a natural number $q\le |\mathfrak{R}|$ and a function $g$ such that, for some $\mathbf{\sigma }\in {\mathfrak{R}}^q$, $V=g(\mathbf{\sigma })$ (valuations depend functionally on subsystem valuations)

Key Characteristics

• Non-uniqueness: The supersystem of a given family is not unique - there are infinitely many ways to construct a supersystem
• Functional dependencies: Supersystem properties arise from (but are not reducible to) subsystem properties
• Aggregation: The supersystem encompasses the underlying sets and elements of all constituent subsystems
• Emergence-permitting: Multiple distinct macroscopic descriptions can be compatible with a single microscopic configuration

Inverse Problem

The definition of supersystem addresses the inverse problem: while one can use knowledge about a system to infer a subsystem’s characterization, when only the latter is known it is much harder to completely describe its supersystems.

The functions that define a subsystem from a given system do not necessarily have inverses. When we define a subsystem, we do not necessarily define how to obtain a corresponding supersystem (which is not even unique).

Non-Uniqueness and Emergence

The non-uniqueness of supersystems is not a defect but a feature of the formalism. It captures a fundamental asymmetry in the relationship between scales:

Emergence, understood as the appearance of novel organizational principles at coarser scales, permits multiple distinct macroscopic descriptions to be compatible with a single microscopic configuration. This reflects the essential underdetermination of coarse-grained descriptions by fine-grained dynamics.

Relationship to Other Concepts

The supersystem concept is the dual of the subsystem concept, but with important asymmetries:

• Subsystem → Supersystem: Many-to-many (non-unique supersystems)
• Supersystem → Subsystem: One-to-many (unique supersystem implies multiple subsystems)

When a supersystem additionally satisfies the enclose relation, it becomes part of a nested system hierarchy. The supersystem concept is also related to superlevel systems, though the latter focuses specifically on level-transitions in multi-scale hierarchies.

Examples

Gas Regions Example

Consider regions of a gas:

• Subsystems: Individual regions characterized by molecular positions and velocities
• Supersystem: The entire gas characterized by temperature fields and pressure gradients

The macroscopic properties (temperature, pressure) are functions of the mesoscopic dynamics (molecular velocities), but multiple different microscopic configurations can yield the same macroscopic state.

Biological Organization

For a multicellular organism:

• Subsystems: Individual cells with intracellular dynamics
• Supersystem: Tissue with intercellular signaling and collective behavior

The tissue exhibits emergent properties (mechanical strength, coordinated response to stimuli) that are functionally dependent on but not reducible to individual cell properties.

Organizational Systems

For a company:

• Subsystems: Individual departments (Engineering, Sales, etc.)
• Supersystem: The company as a whole

The company-level strategy and performance depend on department operations, but the same departments could be combined in multiple different organizational structures.

Philosophical Significance

The supersystem formalism captures three crucial insights:

1. Scale asymmetry: Coarse-graining is many-to-one, but fine-graining is one-to-many
2. Functional dependence: Higher-level properties are functions of lower-level properties
3. Emergence: Novel organizational principles can appear at coarser scales

Key References

Emergent Nested Systems

C. Walloth (2016) View in Semantic Scholar DOI: 10.1007/978-3-319-27550-5

Explores how emergent phenomena arise in nested system hierarchies, where subsystems and supersystems interact across multiple scales.

General systems theory: Mathematical foundations

Mesarovic & Takahara (1976) DOI: 10.1016/0001-8708(76)90196-1

Provides foundational mathematical treatment of system composition and decomposition.

Related Concepts

• subsystem - The dual concept focusing on decomposition
• nested-system - Systems that are both subsystems and supersystems
• parthood - The enclose relation combining subsystem and membership
• superlevel-system - Level-transition operation in hierarchies
• hierarchy - Multi-level organizational structures

Bibliography Keys

• Walloth2016
• Mesarovic1976
• Backlund2000
• Mesarovic1970