Look: Simple Stochastic Relations Are Just, Well, Simple
Simple systems cannot decomposed further. Algebraically, simple systems have only isomorphisms as epis. We characterize simple stochastic relations through different forms of bisimulations for the case that the underlying spaces are Polish, and analytic, respectively. This requires a closer investigation of bisimulations, congruences and their mutual relationship. We provide a complete characterization of simple stochastic relations for analytic spaces.
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