Mathematica Bohemica

Mathematica Bohemica, first online, pp. 1-37

An example Ginsburg said in 1984 he was "unable to find" and a forbidden subposet characterization of subsets of regular posets

Jonathan David Farley

Received March 12, 2023. Published online December 10, 2024.

Abstract: In 1984, Ginsburg wrote, "We have been unable to find an example of an ordered set $P$ having the properties of [being complete, densely ordered, with no antichain other than $\{0\}$ and $\{1\}$ that is a cutset] and in which all antichains are countable." In this very brief note, such an example is shown. Posets that can be embedded in regular posets are characterized as posets that do not contain $\omega\times\{0,1\}$ or its dual as a subposet. Any such poset $P$ can be embedded in a regular poset that can be embedded in any other regular poset containing $P$.

Keywords: regular poset; (minimal) cutset; (maximal) chain; (maximal) antichain; lexicographic sum; complete lattice

Classification MSC: 06A06

DOI: 10.21136/MB.2024.0039-23

PDF available at: Institute of Mathematics CAS

References:
[1] G. Birkhoff: Lattice Theory. AMS Colloquium Publications 25. AMS, Providence (1967). DOI 10.1090/coll/025 | MR 0227053 | Zbl 0153.02501
[2] B. A. Davey, H. A. Priestley: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002). DOI 10.1017/CBO9780511809088 | MR 1902334 | Zbl 1002.06001
[3] M. H. El-Zahar, N. Zaguia: Antichains and cutsets. Combinatorics and Ordered Sets Contemporary Mathematics 57. AMS, Providence (1986), 227-261. DOI 10.1090/conm/057 | MR 0856238 | Zbl 0595.06003
[4] J. D. Farley: Perfect sequences of chain-complete posets. Discrete Math. 167/168 (1997), 271-296. DOI 10.1016/S0012-365X(96)00234-8 | MR 1446751 | Zbl 0873.06002
[5] J. D. Farley: Cancel culture: The search for universally cancellable exponents of posets. Categ. Gen. Algebr. Struct. Appl. 19 (2023), 1-27. DOI 10.48308/CGASA.19.1.1 | MR 4650120 | Zbl 7898816
[6] J. Ginsburg: Compactness and subsets of ordered sets that meet all maximal chains. Order 1 (1984), 147-157. DOI 10.1007/BF00565650 | MR 0764322 | Zbl 0555.06001
[7] P. A. Grillet: Maximal chains and antichains. Fundam. Math. 65 (1969), 157-167. DOI 10.4064/fm-65-2-157-167 | MR 0244112 | Zbl 0191.00601
[8] D. Higgs: A companion to Grillet's theorem on maximal chains and antichains. Order 1 (1985), 371-375. DOI 10.1007/BF00582742 | MR 0787548 | Zbl 0571.06001
[9] B. Y. Li: The PT-order, minimal cutsets and Menger property. Order 6 (1989), 59-68. DOI 10.1007/BF00341637 | MR 1020457 | Zbl 0693.06001
[10] R. Maltby: When is every minimal cutset an antichain? Acta Sci. Math. 59 (1994), 381-403. MR 1317160 | Zbl 0821.06001
[11] G. Markowsky: Chain-complete posets and directed sets with applications. Algebra Univers. 6 (1976), 53-68. DOI 10.1007/BF02485815 | MR 0398913 | Zbl 0332.06001
[12] I. Rival, N. Zaguia: Effective constructions of cutsets for finite and infinite ordered sets. Acta Sci. Math. 51 (1987), 191-207. MR 0911570 | Zbl 0629.06004
[13] B. Schröder: Ordered Sets: An Introduction with Connections from Combinatorics to Topology. Birkhäuser, Basel (2016). DOI 10.1007/978-3-319-29788-0 | MR 3469976 | Zbl 1414.06001
[14] E. Slatinský: Die arithmetische Operation der Summe. Arch. Math., Brno 20 (1984), 9-20. (In German.) MR 0785042 | Zbl 0554.06007
[15] K. Weihrauch, U. Schreiber: Embedding metric spaces into CPO's. Theor. Comput. Sci. 16 (1981), 5-24. DOI 10.1016/0304-3975(81)90027-X | MR 0632667 | Zbl 0485.68040

Affiliations: Jonathan David Farley, Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA, e-mail: lattice.theory@gmail.com

[List of online first articles] [Contents of Mathematica Bohemica] [Full text of the older issues of Mathematica Bohemica at DML-CZ]

PDF available at:

Institute of Mathematics CAS
free access