40 (2006), 3–13 Kazimierz SWIRYDOWICZ UPPER PART OF THE LATTICE OF EXTENSIONS OF THE POSITIVE RELEVANT

A b s t r a c t. In this paper it is proved that the interval [R +, L(2 +)] of the lattice of extensions of the positive (i.e. negationless) relevant logic R + has exactly two co-atoms (L(2 +) denotes here the only Post-complete extension of R +). One of these two co-atoms is the only maximal extension of R + which satisfies the relevance property: if A → B is a theorem then A and B have a common variable. A result of this kind for the relevant logic R was presented in Swirydowicz [1999]. 1. Preliminaries. R +-algebras 1. Let a set of propositional variables p, q, r,... be given and let F be the set of propositional formulae built up from propositional variables by means of the connectives: → (implication), ∧ (conjunction), ∨ and (disjunction)