State-transition categories. (English) Zbl 0564.93002
Consider the state-transition system (S,X,\(\delta)\), with \(\delta\) : \(S\times X\to S\) (S a non-empty set, X the underlying set of a monoid) satisfying: \(\delta\) (s,-) is a homomorphism in the obvious sense. Associate to it the category \(C_{SX}\) whose set of objects is \({\mathcal P}(S)\), the power-set of S and where an arrow \(A\to B\) is the obvious restriction of \(\delta\) (-,x) to domain A and codomain B (A,B subsets of S). A necessary condition is given in functorial terms that a morphism of transition-systems be what one would expect. Then knowledge-functions are introduced and characterized in terms of the \(L_ 2\)-fuzzy category (in Goguen’s sense) on \(C_{SX}\). There are also considerations of how to interpret ’universality’ (universal elements?) in these terms.
Reviewer: M.Eytan
MSC:
93A10 | General systems |
18A20 | Epimorphisms, monomorphisms, special classes of morphisms, null morphisms |
18B15 | Embedding theorems, universal categories |
03E72 | Theory of fuzzy sets, etc. |
08A99 | Algebraic structures |
94D05 | Fuzzy sets and logic (in connection with information, communication, or circuits theory) |
References:
[1] | Goguen, J. A., \(L\)-fuzzy sets, J. Math. Anal. Appl., 18, 145-174 (1967) · Zbl 0145.24404 |
[2] | MacLane, S., Categories for the Working Mathematician (1971), Springer: Springer Berlin · Zbl 0232.18001 |
[3] | Mesarovic, M. D.; Takahara, Yasuhiko, General Systems Theory: Mathematical Foundations (1975), Academic Press: Academic Press New York · Zbl 0328.93002 |
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