Sequent-systems and groupoid models. II
This is the sequel to part I [ibid. 47, 353-385 (1988; Zbl 0671.03018)]. Generalizing relevant model structures (of Urquhart, Fine, Routley, Meyer and Maksimova) the author introduces groupoid frames. Such a frame is a quadruple \(<W,\cdot,\cap,1>\) in which \(<W,\cdot >\) is a groupoid, \(<W,\cap >\) is a semilattice, \(1\in W\), and \(\cdot\) is two-side distributive w.r.t. \(\cap\). The minimal system L introduced in part I is shown to be complete w.r.t. the class of all groupoid frames. Correspondence and completeness results are sketched also for some extensions of L. In particular, Johansson logic is determined by those groupoid frames in which \(<W,\cdot,1>\) is a commutative monoid and \(1\leq x\), \(x\cdot x\leq x\) for any x. Semantics of this type for disjunctionless fragments of extensions of L is presented as well. Also semantics for some extensions of the negationless fragment of the relevant system R is discussed.
scientific article; zbMATH DE number 4127176