Weakening Relation Algebras and FL$$^2$$-algebras

Abstract

FL -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 -algebras. We prove that the congruences of FL -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 , the double division conucleus image 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 of subalgebras of is a discriminator variety of cyclic involutive GBI-algebras that includes RA. We investigate 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 , and we prove that the identity holds only in trivial members of .

Representable and diagonally representable weakening relation algebras

“…The closely related algebras Wk(X) are defined as the expansions of wk(X) by the Heyting implication R → S = {(x, y) | ∀u, v(u ≤ x & y ≤ v & uRv ⇒ uSv)}. The SP-closure of these algebras is denoted by RWkRA and has been studied in [3], [4], [9], [20], [21]. It is a discriminator variety that has RRA of representable relation algebras as a proper subvariety.…”