Abstract
We consider all the full dualities for the class of finite bounded distributive lattices that are based on the three-element chain 3. Under a natural quasi-order, these full dualities form a doubly algebraic lattice \({\mathcal{F}_{\underline{3}}}\). Using Priestley duality, we establish a correspondence between the elements of \({\mathcal{F}_{\underline{3}}}\) and special enriched ordered sets, which we call ‘coloured ordered sets’. We can then use combinatorial arguments to show that the lattice \({\mathcal{F}_{\underline{3}}}\) has cardinality \({2^{\aleph_{0}}}\) and is non-modular. This is the first investigation into the structure of an infinite lattice of finite-level full dualities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Clark, D.M., Davey, B.A.: Natural Dualities for the Working Algebraist. Cambridge University Press (1998)
Clark D.M., Davey B.A., Willard R.: Not every full duality is strong!. Algebra Universalis 57, 375–381 (2007)
Davey, B.A.: Dualisability in general and endodualisability in particular. In: Ursini, A., Aglianó, P. (eds.) Logic and Algebra (Pontignano, 1994). Lecture Notes in Pure and Appl. Math., vol. 180, pp. 437–455. Dekker (1996)
Davey B.A., Haviar M.: A schizophrenic operation which aids the efficient transfer of strong dualities. Houston J. Math. 26, 215–222 (2000)
Davey B.A., Haviar M.: Applications of Priestley duality in transferring optimal dualities. Studia Logica 78, 213–236 (2004)
Davey B.A., Haviar M., Niven T.: When is a full duality strong?. Houston J. Math. 33, 1–22 (2007)
Davey B.A., Haviar M., Priestley H.A.: The syntax and semantics of entailment in duality theory. J. Symbolic Logic 60, 1087–1114 (1995)
Davey B.A., Haviar M., Priestley H.A.: Endoprimal distributive lattices are endodualisable. Algebra Universalis 34, 444–453 (1995)
Davey B.A., Haviar M., Priestley H.A.: Kleene algebras: a case-study of clones and dualities from endomorphisms. Acta Sci. Math. (Szeged) 67, 77–103 (2001)
Davey B.A., Haviar M., Willard R.: Full does not imply strong, does it?. Algebra Universalis 54, 1–22 (2005)
Davey B.A., Haviar M., Willard R.: Structural entailment. Algebra Universalis 54, 397–416 (2005)
Davey, B.A., Pitkethly, J.G., Willard, R.: The lattice of alter egos (2009, preprint). http://www.latrobe.edu.au/mathstats/staff/davey
Davey B.A., Priestley H.A.: Optimal natural dualities. Trans. Amer. Math. Soc. 338, 655–677 (1993)
Davey B.A., Priestley H.A.: Optimal natural dualities for varieties of Heyting algebras. Studia Logica 56, 67–96 (1996)
Davey, B.A., Werner, H.: Dualities and equivalences for varieties of algebras. In: Huhn, A.P., Schmidt, E.T. (eds.) Contributions to Lattice Theory (Szeged, 1980). Colloq. Math. Soc. János Bolyai, vol. 33, pp. 101–275. North-Holland (1983)
Priestley H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)
Priestley H.A.: Ordered topological spaces and the representation of distributive lattices. Proc. London Math. Soc. (3) 24, 507–530 (1972)
Sprenger, M.: Algebra WorkBench. http://www.algebraworkbench.net
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by K. Kearnes.
The second author was supported by Slovak grants APVV-51-009605 and VEGA 1/0485/09, and the third author by ARC Discovery Project Grant DP0556248.
Rights and permissions
About this article
Cite this article
Davey, B.A., Haviar, M. & Pitkethly, J.G. Using coloured ordered sets to study finite-level full dualities. Algebra Univers. 64, 69–100 (2010). https://doi.org/10.1007/s00012-010-0090-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-010-0090-3