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 in representable algebras and give finite axiomatizations for several notions of validity. These results will be applied in the context of substructural logics.

Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.

We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations.

In this paper articulated in two parts we propose a structural interpretation of Schröder’s work, pointing out his insistence on the priority of a whole in comparison with its parts. The examples are taken from the diverse areas in which Schröder was active, with a particular interest in his project of an absolute algebra. I am regretting for the bad quality of my English, hoping that notwithstanding the reader can grasp at least the fundamental tracts of my reasoning.