Abstract
The representation of partially ordered sets by subsets of some set such that specified joins (meets) are taken to unions (intersections) suggests two categories, that of partially ordered sets with specified joins and meets, and that of sets equipped with suitable collections of subsets, and adjoint contravariant functors between them. This, in turn, induces a duality including, among several others, the two Stone Dualities and that between spatial locales and sober spaces.
Similar content being viewed by others
References
B. Banaschewski (1956) Hüllensystems und Erweiterung von Quasiordnungen, Z. Mathem. Logik Grundl. Mathem. 2, 117–130.
G. Bruns (1962) Darstellungen und Erweiterungen geordneter Mengen, I and II. J. Reine Angew. Mathem. 209, 167–200 and 210, 1–23.
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott (1980) A Compendium of Continuous Lattices, Springer-Verlag, Berlin.
P. T. Johnstone (1983) Stone Spaces, Cambridge University Press.
S. Mac Lane (1971) Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, Berlin.
J. B. Wright, E. G. Wagner, and J. W. Thatcher (1978) A uniform approach to inductive posets and inductive closure, T.C.S. 7, 57–77.
Author information
Authors and Affiliations
Additional information
Communicated by R. Wille
Rights and permissions
About this article
Cite this article
Banaschewski, B., Bruns, G. The fundamental duality of partially ordered sets. Order 5, 61–74 (1988). https://doi.org/10.1007/BF00143898
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00143898