Transitivity and Difunctionality of Bisimulations

Abstract : Bisimilarity and observational equivalence are notions that agree in many classical models of coalgebras, such as e.g. Kripke structures. In the general category $Set_{F}$-coalgebras these notions may, however, diverge. In many cases, observational equivalence, being transitive, turns out to be more useful.In this paper, we shall investigate the role of transitivity for the largest bisimulation of a coalgebra. Passing to relations between two coalgebras, we choose difunctionality as generalization of transitivity. Since in $Set_{F}$ bisimulations are known to coincide with $\bar{F}-$ simulations, we are led to study the notion of $L-$ similarity, where L is a relation lifting.