Model Completions in Modal Logic

Abstract

In this section, we shall use Theorem 5.49 in order to obtain negative results, i.e. in order to show that a model completion does not exist for many equational theories containing K4. Let n be the n-elements linearly ordered reflexive chain, that is n is the frame consisting of the set {1, 2, ..., n} ordered by the natural order ≤. n* is the frame obtained by n by replacing 1 by a two-elements cluster, that is n* has {1⁺, 1^-, 2, ..., n} as underlying set and the order is defined by cases by p ≤ q iff i) either p = 1⁺ or ii) p = 1^- or iii) (p, q ≤ 2 and p ≤ q).