Abstract
A lattice L is constructed with the property that every interval has finite height, but there exists no strictly order-preserving map from L to Z. A 1979 problem of Erné (posed at the 1981 Banff Conference on Ordered Sets) is thus solved. It is also shown that if a poset P has no uncountable antichains, then it admits a strictly order-preserving map into Z if and only if every interval has finite height.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Daykin, D. E. and Daykin, M. W. (1995) Order preserving maps and linear extensions of a finite poset, SIAM J. Algebraic Discrete Meth. 6, 738–748.
Erné, M. (1979) Order and Topology Lecture Notes, Universität Hannover, Hannover, Germany.
Ginsburg, J., Rival, I. and Sands, B. (1986) Antichains and finite sets that meet all maximal chains, Canad. J. Math. 38, 619–632.
Kale, K. A. (1996) Monotone functions from posets onto ω, Bull. London Math. Soc. 28, 1–3.
Rival, I. (ed.) (1982) Ordered Sets, D. Reidel, Dordrecht.
Rival, I. (ed.) (1985) Graphs and Order: The Role of Graphs in the Theory of Ordered Sets and Its Applications, D. Reidel, Dordrecht.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Farley, J.D., Schröder, B.S.W. Strictly Order-Preserving Maps into Z, II. A 1979 Problem of Erné. Order 18, 381–385 (2001). https://doi.org/10.1023/A:1013929314438
Issue Date:
DOI: https://doi.org/10.1023/A:1013929314438