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,
*S*_{1} ⊂ (*X*_{1}
× *Z*_{21}) × (*Y*_{1}
× *Z*_{12}) and *S*_{2}
⊂ (*X*_{2} × *Z*_{12})
× (*Y*_{2} ×
*Z*_{21}), the system *S*
= *S*_{1}∗ *S*_{2}
⊂ (*X*_{1} × *X*_{2})
× (*Y*_{1 } ×
*Y*_{2}) is given by

S:= { ((*x*_{1}, *x*_{2}),
(*y*_{1}, *y*_{2}))
| ∃ *z*_{1} ∈ *Z*_{12},
*z*_{2} ∈ *Z*_{21},
such that ((*x*_{1}, *z*_{2}),
(*y*_{1},
*z*_{1})) ∈ *S*_{1},
((*x*_{2}, *z*_{1}),
(*y*_{2}, *z*_{2}))
∈ *S*_{2}} .

* __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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