Representation of Residuated Semigroups in Some Algebras of Relations (The Method of Canonical Models)

We prove some theorems on representation of residuated semigroups and monoids in algebras of binary relations. One of them has been proved in Andréka and Mikulás [3], and the others are new, though also closely related to some results in [3]. The main novelty of this paper is a simple method of proof, based upon a construction of canonical models for Lambek calculi by means of labeled formulas, whereas [3] uses graph-theoretic constructions. We handle labeled formulas in a way similar to Kurtonina [15] and the second author [14], but we make no use of Labeled Deductive Systems, which is an essential simplification.