  In the present paper we consider modal propositional logic and look for the
constraints that are imposed to the propositions of the special type $\Box a$
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 $\Box a$ can be derived through a binomial
ideal containing a toric ideal and we give sufficient conditions under which
the toric ideal fully describes the constraints.
