Abstract In the present paper, we consider modal propositional logic and look for the constraints that are imposed to the propositions of the special type $\operatorname{\Box } a$ by the structure of the relevant finite Kripke frame. We translate the usual language of modal propositional logic in terms of notions of commutative algebra, namely polynomial rings, ideals and bases of ideals. We use extensively the perspective obtained in previous works in algebraic statistics. We prove that the constraints on $\operatorname{\Box } a$ can be derived through a binomial ideal containing a toric ideal and we give sufficient conditions under which the toric ideal, together with the fact that the truth values are in $\left \{0,1\right \} $, fully describes the constraints.

Abstract It is shown that the relations of recursive isomorphism on countable trees, groups, Boolean algebras, fields and total orderings are universal countable Borel equivalence relations, thus providing a countable analogue of the Borel completeness of the isomorphism relations on these same classes.