Coalgebraic Simulations and Congruences

Abstract : In a recent article Gorín and Schröder study $\lambda$-simulations of coalgebras and relate them to preservation of positive formulae. Their main results assume that $\lambda$ is a set of monotonic predicate liftings and their proofs are set-theoretical. We give a different definition of simulation, called strong simulation, which has several advantages: Our notion agrees with that of in the presence of monotonicity, but it has the advantage, that it allows diagrammatic reasoning, so several results from the mentioned paper can be obtained by simple diagram chases. We clarify the role of $\lambda$-monotonicity by showing the equivalence of
- $\lambda$ is monotonic
- every simulation is strong
- every bisimulation is a (strong) simulation
- every F-congruence is a (strong) simulation.
We relate the notion to bisimulations and F-congruences - which are defined as pullbacks of homomorphisms. We show that
- if $\lambda$ is a separating set, then each difunctional strong simulation is an $F$-congruence,
- if $\lambda$ is monotonic, then the converse is true: if each difunctional strong simulation is an $F$-congruence, then $\lambda$ is separating.
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H. Gumm, Mehdi Zarrad. Coalgebraic Simulations and Congruences. 12th International Workshop on Coalgebraic Methods in Computer Science (CMCS), Apr 2014, Grenoble, France. pp.118-134, ⟨10.1007/978-3-662-44124-4_7⟩. ⟨hal-01408755⟩

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