The structure of generalized BI-algebras and weakening relation algebras

• Published in: Algebra universalis Vol. 81; no. 3
• Main Authors: Galatos, Nikolaos, Jipsen, Peter
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-08-2020; Springer Nature B.V
• Subjects: Algebra; Boolean algebra; Congruences; Construction; Lattices (mathematics); Mathematics; Mathematics and Statistics; Operators (mathematics); Relation algebra; Substructural logic; 08B15; Residuated lattice; Weakening relation; 06F05; Bunched implication algebra; Discriminator; 03B47; 03G10
• ISSN: 0002-5240; 1420-8911
• DOI: 10.1007/s00012-020-00663-9

Generalized bunched implication algebras (GBI-algebras) are defined as residuated lattices with a Heyting implication, and are positioned between Boolean algebras with operators and lattices with operators. We characterize congruences on GBI-algebras by filters that are closed under Gumm–Ursini terms, and for involutive GBI-algebras these terms simplify to a dual version of the congruence term for relation algebras together with two more terms. We prove that representable weakening relation algebras form a variety of cyclic involutive GBI-algebras, denoted by RWkRA, containing the variety of representable relation algebras. We describe a double-division conucleus construction on residuated lattices and on (cyclic involutive) GBI-algebras and show that it generalizes Comer’s double coset construction for relation algebras. Also, we explore how the double-division conucleus construction interacts with other class operators and in particular with variety generation. We focus on the fact that it preserves a special discriminator term, thus yielding interesting discriminator varieties of GBI-algebras, including RWkRA. To illustrate the generality of the variety of weakening relation algebras, we prove that all distributive lattice-ordered pregroups and hence all lattice-ordered groups embed, as residuated lattices, into representable weakening relation algebras on chains. Moreover, every representable weakening relation algebra is embedded in the algebra of all residuated maps on a doubly-algebraic distributive lattice. We give a number of other instructive examples that show how the double-division conucleus illuminates the structure of distributive involutive residuated lattices and GBI-algebras.