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 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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jun 2016