It is well-known that a Boolean algebra (B, +, ., ‐) may be defined as an algebraic system with at least two elements such that (for all x, y, z ε B):

These axioms or equations are not independent, in the sense that some of them are logical consequences of the others. B. A. Bernstein [1] thought that the first three and their duals form an independent dual-symmetric definition of a Boolean algebra, but R. Montague and J. Tarski [3] proved later that B1 (or B̅1) follows from B2, B3, B̅1, B̅2, B̅3 (from B1, B2, B3, B̅2, B̅3).

We consider the results of adding to a basic modal system T0 the axioms G1. CLpp; Pn. CLnpLn+1p; Bn. CpLnMp, where n ≧ 11, in all combinations. The method of Meredith's [7] will be extended to get models of these systems in lower predicate calculus (LPC) with a constant binary relation, U. Most of the results were already obtained in [1]–[6], though systems as in (i) and (ii) below were not investigated, except that S40 in (ii) was mentioned in [1]. However some repetition may be excused in view of the simplicity with which the results are obtained by the present method.