System Theory and Physical Systems

In General
$ System: A relation S ⊂ X × Y for some two sets X = ×_{i ∈ I} V_i (input set) and Y = ×_{i ∈ I} W_i (output set).
$ State set and response function: A pair C, R with R: (C × X) → Y and

(x, y) ∈ S  iff  there exists  c ∈ C  such that  R(c, x) = y .

* Conditions: Any system has a C and R; A state set can be defined, e.g., by the set of functions

C:= {f_C | f_c: X → Y, f_c ⊂ S} .

$ Composition of systems: Given two systems, S₁ ⊂ (X₁ × Z₂₁) × (Y₁ × Z₁₂) and S₂ ⊂ (X₂ × Z₁₂) × (Y₂ × Z₂₁), the system S = S₁∗ S₂ ⊂ (X₁ × X₂) × (Y₁ × Y₂) is given by

S:= { ((x₁, x₂), (y₁, y₂)) | ∃ z₁ ∈ Z₁₂, z₂ ∈ Z₂₁, such that ((x₁, z₂), (y₁, z₁)) ∈ S₁, ((x₂, z₁), (y₂, z₂)) ∈ S₂} .

* Symmetry: Given a group G acting on X × Y, a system S is G-symmetric iff

for all g ∈ G, x ∈ X, y ∈ Y,   (gx, gy) ∈ S iff (x, y) ∈ S .

@ References: Smullyan 61; Mesarović & Takahara 75.
> Online resources: see Wikipedia page.

Systems in Physics > s.a. classical systems; Emergent Systems; Isolated Systems; Open Systems; state of a system.
* Ingredients: A physical system has a state, including internal and external correlations, and internal and external interactions.
@ General references: Szabó IJTP(86) [and elementary objects]; Aerts & Pulmannová JMP(06)-a0811 [state property systems]; Lee & Hoban EPTCS(16)-a1606 [information content, communication complexity].
@ Dimensionality of a system: Wolf & Pérez-García PRL(09)-a0902 [quantum systems, from evolution]; Gallego et al PRL(10)-a1010, Hendrych et al nPhys(12)-a1111 [tests].
@ In quantum theory: Dugić & Jeknić IJTP(06)qp/05 [and decoherence theory]; Dugić & Jeknić-Dugić IJTP(08)qp/06 [information-theoretic arguments]; Seidewitz FP(11)-a1002 [systems, subsystems, and their interactions]; > s.a. quantum foundations [ontology]; quantum systems.

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