Abstract

In the following we will use the well-known correspondence between modal logics and varieties of modal algebras in our investigation of the function which assigns to a modal logic its degree of incompleteness. A modal algebra is an algebra A = where is a Boolean algebra and is a unary operation satisfying 1 = 1 and = x y ; is called a modal operator. A variety of algebras is a class of algebras closed under the operations of forming homomorphic images, subalgebras as and direct products, and if K is a class of algebras then V denotes the smallest variety containing K. The variety of modal algebras is denoted by M, the subvariety of M dened by the equation x x = x by MR and the subvariety of M dened by the equation x n = x n1 , n a natural number, by Mn . Here x 0 = x, x n = , n a natural number. We write for the lattice of subvarieties of a variety K. If K, K 0 are varieties such that K $ K 0 but for no variety K 00 K $ K 00 $ K 0 then we say that K 0 is a cover of K. The smallest normal modal logic { containing the axiom ! and closed under the inference rules of modus ponens, substitution and necessitation { will be denoted by K; the lattice of modal logics which are extensions of K by