Abstract
We study the pointed lattice subreducts of varieties of residuated lattices (RLs) and commutative residuated lattices (CRLs), i.e. lattice subreducts expanded by the constant \(\textsf{1}\) denoting the multiplicative unit. Given any positive universal class of pointed lattices \(\textsf{K}\) satisfying a certain equation, we describe the pointed lattice subreducts of semi-\(\textsf{K}\) and of pre-\(\textsf{K}\) RLs and CRLs. The quasivariety of semi-prime-pointed lattices generated by pointed lattices with a join prime constant \(\textsf{1}\) plays an important role here. In particular, the pointed lattice reducts of integral (semiconic) RLs and CRLs are precisely the integral (semiconic) semi-prime-pointed lattices. We also describe the pointed lattice subreducts of integral cancellative CRLs, proving in particular that every lattice is a subreduct of some integral cancellative CRL. This resolves an open problem about cancellative CRLs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adaricheva, K., Maróti, M., McKenzie, R., Nation, J.B., Zenk, E.R.: The Jónsson-Kiefer property. Stud. Logica. 83(1), 111–131 (2006)
Bahls, P., Cole, J., Galatos, N., Jipsen, P., Tsinakis, C.: Cancellative residuated lattices. Algebra Universalis 50(1), 83–106 (2003)
Baker, Kirby A., Hales, Alfred W.: From a lattice to its ideal lattice. Algebra Universalis 4, 250–258 (1974)
Bergman, C.: Universal algebra: fundamentals and selected topics, vol. 151 of Studies in Logic and the Foundations of Mathematics. Elsevier, (2007)
Blount, K., Tsinakis, C.: The structure of residuated lattices. Internat. J. Algebra Comput. 13(4), 437–461 (2003)
Burris, S., Sankappanavar, H. P.: A course in universal algebra, Millenium Edition. (2012). Available at https://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html
Crawley, P., Dilworth, R.P.: Algebraic theory of lattices. Prentice-Hall (1973)
Czelakowski, J., Dziobiak, W.: Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class. Algebra Universalis 27, 128–149 (1990)
Fussner, W., Galatos, N.: Semiconic idempotent logic I: structure and local deduction theorems. https://arxiv.org/abs/2208.09724v2
Galatos, N., Horčík, R.: Cayley’s and Holland’s theorems for idempotent semirings and their applications to residuated lattices. Semigroup Forum 87(3), 569–589 (2013)
Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated lattices: an algebraic glimpse at substructural logics, vol. 151 of Studies in Logic and the Foundations of Mathematics. Elsevier (2007)
Horčík, R.: Finite embeddability property for residuated lattices via regular languages. In Nikolaos Galatos and Kazushige Terui, editors, Hiroakira Ono on Substructural Logics, vol. 23 of Outstanding Contributions to Logic, pages 273–298. Springer (2022)
Hsieh, A.N., Raftery, J.G.: Semiconic idempotent residuated structures. Algebra Universalis 61, 413–430 (2009)
Jónsson, B.: Congruence distributive varieties. Mathematica Japonica, 42(2):353–401, 95
Jónsson, B., Kiefer, J.E.: Finite sublattices of a free lattice. Can. J. Math. 14, 487–497 (1962)
Metcalfe, G., Paoli, F., Tsinakis, C.: Residuated structures in algebra and logic, vol. 277 of Mathematical Surveys and Monographs. American Mathematical Society (2023)
Montagna, F., Tsinakis, C.: Ordered groups with a conucleus. J. Pure Appl. Algebra 214(1), 71–88 (2010)
Stern, M.: Semimodular lattices: Theory and applications. Number 73 in Encyclopedia of Mathematics and its Applications. Cambridge University Press (1999)
Young, W.: Heyting algebras as intervals of commutative, cancellative residuated lattices. Unpublished manuscript (2014)
Acknowledgements
The author is grateful to an anonymous referee for their careful reading of the manuscript and a number of helpful suggestions, and to Nick Galatos and Peter Jipsen for their comments on an earlier draft of Section 4.
Funding
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work was funded by the grant 2021 BP 00212 of the grant agency AGAUR of the Generalitat de Catalunya.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Semidistributivity semidistributive ultraproduct quasivariety subdirect subdirectly multiplication multiplicative semiseparated intersection idempotent subalgebra homomorphism homomorphisms isomorphism isomorphisms.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Přenosil, A. Pointed Lattice Subreducts of Varieties of Residuated Lattices. Order (2024). https://doi.org/10.1007/s11083-024-09671-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11083-024-09671-z