Bisimulation and Hidden Algebra

A hidden algebra is a special case of coalgebra. A hidden congruence on a hidden algebra corresponds to a bisimulation equivalence on the corresponding coalgebra. The paper generalizes the notion of hidden congruence to that of hidden bisimulation between two different hidden algebras. We first define hidden bisimulation between two hidden algebras having the same signature. A hidden bisimulation is in fact a bisimulation between the corresponding coalgebras. We then define hidden simulation between two hidden algebras having different signatures related by a vertical signature morphism. We prefer to call this relation simulation because it is unidirectional, due to the signature morphism. For the last case, the relationship between simulation and refinement is discussed.