State-transition categories (Q1058492)

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.