Author:
Fazio D.,Ledda A.,Paoli F.
Abstract
AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$
ℓ
-groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$
ℓ
-groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$
ℓ
-groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$
ℓ
-groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.
Publisher
Springer Science and Business Media LLC
Subject
History and Philosophy of Science,Logic
Reference53 articles.
1. Anderson, A.R., and N.D. Belnap, Entailment, vol. 1, Princeton University Press, Princeton, 1975.
2. Aglianò, P., and A. Ursini, On subtractive varieties II: General properties, Algebra Universalis 36:222–259, 1996.
3. Banaschewski, B., Hüllensysteme und Erweiterung von Quasi-Ordnungen, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 2:117–130, 1956.
4. Beran, L., Central and exchange properties of orthomodular lattices, Mathematische Nachrichten 97:247–251, 1980.
5. Beran, L., Orthomodular Lattices: Algebraic Approach, Reidel, Dordrecht, 1985.
Cited by
3 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献