The Lattice of Congruences of a Finite Linear Frame

Let F = 〈F,R〉 be a finite Kripke frame. A congruence of F is a bisimulation of F that is also an equivalence relation on F. The set of all congruences of F is a lattice under the inclusion ordering. In this article we investigate this lattice in the case that F is a finite linear frame. We give concrete descriptions of the join and meet of two congruences with a nontrivial upper bound. Through these descriptions we show that for every nontrivial congruence ρ, the interval [IdF, ρ] embeds into the lattice of divisors of a suitable positive integer. We also prove that any two congruences with a nontrivial upper bound permute.