The structure of generalized BI-algebras and weakening relation algebras
Autor: | Peter Jipsen, Nikolaos Galatos |
---|---|
Rok vydání: | 2020 |
Předmět: |
Pure mathematics
Algebra and Number Theory Double coset 010102 general mathematics Structure (category theory) Distributive lattice 0102 computer and information sciences Congruence relation Relation algebra 01 natural sciences Distributive property 010201 computation theory & mathematics 0101 mathematics Residuated lattice Variety (universal algebra) Mathematics |
Zdroj: | Algebra universalis. 81 |
ISSN: | 1420-8911 0002-5240 |
DOI: | 10.1007/s00012-020-00663-9 |
Popis: | 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. |
Databáze: | OpenAIRE |
Externí odkaz: |