Abstract
A subresiduated lattice is a pair (A, D), where A is a bounded distributive lattice, D is a bounded sublattice of A and for every \(a,b\in A\) there is \(c\in D\) such that for all \(d\in D\), \(d\wedge a\le b\) if and only if \(d\le c\). This c is denoted by \(a\rightarrow b\). This pair can be regarded as an algebra \(\left<A,\wedge ,\vee ,\rightarrow ,0,1\right>\) of type (2, 2, 2, 0, 0) where \(D=\{a\in A\mid 1\rightarrow a=a\}\). The class of subresiduated lattices is a variety which properly contains to the variety of Heyting algebras. In this paper we present dual equivalences for the algebraic category of subresiduated lattices. More precisely, we develop a spectral style duality and a bitopological style duality for this algebraic category. Finally we study the connections of these results with a known Priestley style duality for the algebraic category of subresiduated lattices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1974)
Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., Kurz, A.: Bitopological duality for distributive lattices and Heyting algebras. Math. Struct. Comput. Sci. 20(3), 359–393 (2010)
Celani, S., Jansana, R.: A closer look at some subintuitionistic logics. Notre Dame J. Formal Logic 42, 225–255 (2003)
Celani, S., Jansana, R.: Bounded distributive lattices with strict implication. Math. Logic Quart. 51, 219–246 (2005)
Cornish, W.H., On, H.: Priestley’s dual of the category of bounded distributive lattices. Matematički Vesnik 12 27(60), 329–332 (1975)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1994)
Epstein, G., Horn, A.: Logics which are characterized by subresiduated lattices. Math. Logic Quart. 22(1), 199–210 (1976)
Fleisher, I.: Priestley’s duality from Stone’s. Adv. Appl. Math. 25(3), 233–238 (2000)
Priestley, H.A.: Representation of bounded distributive lattice by means of orderer Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)
San Martín H.J.: Compatible operations in some subvarieties of the variety of weak Heyting algebras. In: Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013). Advances in Intelligent Systems Research, pp. 475–480. Atlantis Press (2013)
Stone, M.: Topological representation of distributive lattices and Brouwerian logics. Časopis Pro Pěstování Matematiky a Fysiky 67(1), 1–25 (1938)
Acknowledgements
The authors acknowledge many helpful comments from the anonymous referee, which considerably improved the presentation of this paper. The second author was supported by a doctoral fellowship from Comisión de Investigaciones Científicas (CIC-Argentina).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Presented by N. Galatos.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Celani, S.A., Nagy, A.L. & Martín, H.J.S. Dualities for subresiduated lattices. Algebra Univers. 82, 59 (2021). https://doi.org/10.1007/s00012-021-00752-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00012-021-00752-3