Weakening Relation Algebras and FL²-algebras

Authors

Nikolaos Galatos, University of Denver
Peter Jipsen, Chapman UniversityFollow

Files

Description

FL²-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 FL²-algebras. We prove that the congruences of FL²-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 A, the double division conucleus image p/A/p 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 S(dRA) of subalgebras of {p/A/p∶ A ϵ RA; 1 ≤ p² = p ϵ A} is a discriminator variety of cyclic involutive GBI-algebras that includes RA. We investigate S(dRA) 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 0 ≤ 1, and we prove that the identity 1 ≤ 0 holds only in trivial members of S(dRA).

ISBN

978-3-030-43520-2

Publication Date

4-2020

Publisher

Springer

Keywords

Relation algebras, residuated lattices, bunched implication algebras

Disciplines

Algebra

Comments

In Uli Fahrenberg, Peter Jipsen, and Michael Winter (Eds.), Relational and Algebraic Methods in Computer Science: 18th International Conference, RAMiCS 2020, Palaiseau, France, April 8–11, 2020, Proceedings.

Recommended Citation

N. Galataos and P. Jipsen, Weakening Relation Algebras and FL²-algebras, Relational and Algebraic Methods in Computer Science (Warsaw, 1991), Springer International Publ., vol. 18, RAMiCS, Palaiseau, 2020, pp. 117-133.

Copyright

Springer