On Fragments without Implications of both the Full Lambek Logic and some of its Substructural Extensions

In this paper we study some fragments without implications of the (Hilbert) full Lambek logic HFL and also some fragments without implications of some of the substructural extensions of that logic. To do this, we perform an algebraic analysis of the Gentzen systems defined by the substructural calculi FLσ. Such systems are extensions of the full Lambek calculus FL with the rules codified by a subsequence, σ, of the sequence ewlwlc; where e stands for exchange, wl for left weakening, wr for right weakening, and c for contraction. We prove that these Gentzen systems (in languages without implications) are algebraizable by obtaining their equivalent algebraic semantics. All these classes of algebras are varieties of pointed semilatticed monoids and they can be embedded in their ideal completions. As a consequence of these results, we reveal that the fragments of the Gentzen systems associated with the calculi FLσ are the restrictions of them to the sublanguages considered, and we also reveal that in these languages, the fragments of the external systems associated with FLσ are the external systems associated with the restricted Gentzen systems (i.e., those obtained by restriction of FLσ to the implication-less languages considered). We show that all these external systems without implication have algebraic semantics but they are not algebraizable (and are not even protoalgebraic). Results concerning fragments without implication of intuitionistic logic without contraction were already reported in Bou et al. (2006) and Adillon et al. (2007).