Relation Algebras, Idempotent Semirings and Generalized Bunched Implication Algebras

This paper investigates connections between algebraic structures that are common in theoretical computer science and algebraic logic. Idempotent semirings are the basis of Kleene algebras, relation algebras, residuated lattices and bunched implication algebras. Extending a result of Chajda and Länger, we show that involutive residuated lattices are determined by a pair of dually isomorphic idempotent semirings on the same set, and this result also applies to relation algebras. Generalized bunched implication algebras (GBI-algebras for short) are residuated lattices expanded with a Heyting implication. We construct bounded cyclic involutive GBI-algebras from so-called weakening relations, and prove that the class of weakening relation algebras is not finitely axiomatizable. These algebras play a role similar to representable relation algebras, and we identify a finitely-based variety of cyclic involutive GBI-algebras that includes all weakening relation algebras. We also show that algebras of down-closed sets of partially-ordered groupoids are bounded cyclic involutive GBI-algebras.