Weakening Relation Algebras and FL[Formula: see text]-algebras

Galatos, Nikolaos; Jipsen, Peter

FL[Formula: see text]-algebras are lattice-ordered algebras with two sets of residuated operators. The classes RA of relation algebras and GBI of generalized bunched implication algebras are subvarieties of FL[Formula: see text]-algebras. We prove that the congruences of FL[Formula: see text]-algebras are determined by the congruence class of the respective identity elements, and we characterize the subsets that correspond to this congruence class. For involutive GBI-algebras the characterization simplifies to a form similar to relation algebras. For a positive idempotent element p in a relation algebra [Formula: see text], the double division conucleus image [Formula: see text] is an (abstract) weakening relation algebra, and all representable weakening relation algebras (RWkRAs) are obtained in this way from representable relation algebras (RRAs). The class [Formula: see text] of subalgebras of [Formula: see text] is a discriminator variety of cyclic involutive GBI-algebras that includes RA. We investigate [Formula: see text] to find additional identities that are valid in all RWkRAs. A representable weakening relation algebra is determined by a chain if and only if it satisfies [Formula: see text], and we prove that the identity [Formula: see text] holds only in trivial members of [Formula: see text].