Lower Semilattice-Ordered Residuated Semigroups and Substructural Logics

Abstract

Abstract. We look at lower semilattice-ordered residuated semigroups and, in particular, the representable ones, i.e., those that are isomorphic to algebras of binary relations. We will evaluate expressions (terms, sequents, equations, quasi-equations) in representable algebras and give finite axiomatizations for several notions of validity. These results will be applied in the context of substructural logics.

Relational Models for the Lambek Calculus with Intersection and Unit

“…• For these non-standard models, in Section 3 we prove weak completeness of L Λ ∧01. As a corollary, we obtain a simpler proof of Mikulás' completeness result [Mik15a,Mik15b] for the fragment of L Λ ∧01 without constants. • Strong completeness fails, as proved in Section 4.…”

“…Later on, however, Mikulás [Mik15a,Mik15b] managed to modify the proof of Theorem 1.9 for L Λ ∧, but this modification establishes only weak completeness: Theorem 1.10 (Mikulás 2015). 1 If a sequent (in the language of •, \, /, ∧) is true in all square R-models, then it is derivable in L Λ ∧.…”

“…the class of non-standard models defined in Section 2. A series of potential counterexamples to strong completeness was given by Mikulás [Mik15b].…”

“…Proof. The first part (semantic entailment) is due to Mikulás [Mik15b,Remark 5.3]. We reproduce it here in order to keep our exposition self-contained.…”

“…• Strong completeness fails, as proved in Section 4. The counterexample used in this section was suggested by Mikulás [Mik15b], but without proof. Here we fill this gap.…”