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Simulations of Pawlak machines as fuzzy morphisms of partial algebras. (English) Zbl 0359.18005
In this paper an example of fuzzy theory (in the sense of Arbib and Manes) is given; simulations of Pawlak machines are proved to be fuzzy morphismsovera certain category of unary partial algebras. Let us recall that a Pawlak machine is an ordered pair \((A,f)\) in which \(f\) is a partial mapping of \(A\) into itself, i.e. \((A,f)\) is a partial algebra with one unary operation. A mapping \(\alpha:A\to B\) is said to be a simulation of \((A,f)\) in \((B,g)\) if the two following conditions are fulfilled: \[ (\forall a\in A)(a\in D(f)\text{ iff }\alpha(a)\in D(g))\tag{1} \] \[ (\forall a\in D(f))(\exists k_a\geq1)(\alpha\circ f(a)=g^{k_a}(\alpha(a)))\tag{2} \] where \(D(f)\), \(D(g)\) are domains of \(f,g\) respectively. A usual homomorphism of partial algebras which is also a simulation is called s-homomorphism. Actually, in the article fuzzy theory is not explicitly constructed, but a proof is given that the category of Pawlak machines and s-homomorphisms is a coreflective subcategory of the category of Pawlak machines and simulations. Both categories under consideration have the following set of generators: \(\mathfrak G\{(1,+_1),(2,+_2),\dots,(\mathbb N,+)\}\) where \(n=\{0,\dots,n-1\},D(+_n)=n-\{n-1\},+_n(j)=j+1,\mathbb N=\{0,1,2,\dots\},D(+)=\mathbb N,+(j)=j+1\). If we denote the category of Pawlak machines and simulations Sim, then the coreflection of the object \((A,f)\) is constructed as a certain natural factorization of the object \[ \coprod\limits_{G\in\mathfrak G}\coprod\limits_{\iota\in\langle G_1(A,f)\rangle_{\operatorname{Sim}}}G\times\{\iota\}. \]
Reviewer: Jan Fried
18B20 Categories of machines, automata
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