Representation Theorems for Implication Structures

Professor Rasiowa [HR49] considers implication algebras (A,⇒, V ) such that ⇒ is a binary operation on the universe A and V ∈ A. In particular, there are studied implicative algebras, positive implication algebras and implication algebras; the second kind corresponds to the logic of positive implication, and the third one to the logic of classical implication. In each case, the distinguished element V is interpreted as the equivalence class of all theorems in the appropriate Lindenbaum algebra. Accordingly, the approach cannot be directly applied to logics in which theorems do not form a single equivalence class (e.g. BCI) or there are no theorems at all (e.g. the Lambek calculus); such logics are typical for the world of substructural logics, arising from Gentzen style systems of intuitionistic logic by dropping structural rules (see [3]). Instead of implication algebras in the sense of [HR49], one should consider implication structures (A,⇒,≤) such that (A,≤) is a poset, and ⇒ is a binary operation on A. In the Lindenbaum algebra, ≤ corresponds to the (sequential) consequence relation `, determined by the given logic. Usually, implication structures are obtained as reducts of residuated groupoids (A, ◦,⇒,⇐,≤), where (A,≤) is a poset, and ◦,⇒,⇐ are binary operations on A, fulfilling the equivalences: