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

TL;DR

This paper investigates fragments of the full Lambek logic and its substructural extensions without implications, providing algebraic semantics and analyzing their properties and embeddings.

Contribution

It introduces algebraizable Gentzen systems for implication-less fragments of substructural logics and characterizes their algebraic semantics and structural properties.

Findings

Fragments are algebraizable with pointed semilatticed monoid varieties.

External systems without implications have algebraic semantics but are not algebraizable.

Fragments can be embedded in their ideal completions.

Abstract

In this paper we study some fragments without implications of the (Hilbert) full Lambek logic $\mathbf{HFL}$ 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 $\FL_\sigma$. Such systems are extensions of the full Lambek calculus $\FL$ with the rules codified by a subsequence, $\sigma$, of the sequence $e w_l w_r c$; where $e$ stands for \emph{exchange}, $w_l$ for \emph{left weakening}, $w_r$ for \emph{right weakening}, and $c$ for \emph{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 $\FL_\sigma$ 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 $\FL[\sigma]$ are the external systems associated with the restricted Gentzen systems (i.e., those obtained by restriction of $\FL_\sigma]$ 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): On two fragments with negation and without implication of the logic of residuated lattices. Archive for Mathematical Logic 45(5) and in Adill\'on et al. (2007): On three implication-less fragments of t-norm based fuzzy logics. Fuzzy Sets and Systems 158(23).