Ternary spaces, media, and Chebyshev sets. (English) Zbl 0544.51011

Hedlíková, Jarmila

Czech. Math. J. 33(108), 373-389 (1983).

This paper extends various results on ternary algebras, particularly the reviewer’s in Trans. Am. Math. Soc. 260, 319-362 (1980; Zbl 0446.06007). Some are extended to abstract ternary spaces: sets with a relation abc that is reversible \((abc\Rightarrow cba)\), separated (abc and acb \(\Leftrightarrow b=c)\), and satisfies two patching conditions (if abc and acd then abd and bcd). The prototype is a modular lattice, where abc is defined by \(b=(b\bigwedge a)\bigvee(b\bigwedge c)=(b\bigvee)\bigwedge(b\bigvee c)\) [E. Pitcher and M. F. Smiley, Trans. Am. Math. Soc. 52, 95-114 (1942; Zbl 0060.064)]; reviewer extended to certain ternary algebras characterized as ”modular subsets” of general lattices. The author now calls those I-media and generalizes to (it seems) more ternary algebras, called media. Another class of examples is a simply determined subclass of the join spaces of W. Prenowitz and J. Jantosciak [J. Reine Angew. Math. 257, 100-128 (1972; Zbl 0264.50002)], containing all vector spaces over ordered fields and indeed over fields with a compatible ternary space structure; these last were called ”partially ordered fields” by W. Prenowitz [Amer. Math. Monthly 53, 439-449 (1946; Zbl 0060.324)], but they are not what are now so called. Essentially, the author shows that Jordan-Hölder theory and Chebyshev set theory extend, the descriptive results to ternary spaces, the existential results to media.

Reviewer: J.R.Isbell

MSC:

51G05	Ordered geometries (ordered incidence structures, etc.)
06C10	Semimodular lattices, geometric lattices
20N10	Ternary systems (heaps, semiheaps, heapoids, etc.)

Keywords:

modular subsets of general lattices; abstract ternary spaces; ternary algebras; I-media; partially ordered fields; Jordan-Hölder theory; Chebyshev set theory

Citations:

Zbl 0446.06007; Zbl 0060.064; Zbl 0264.50002; Zbl 0060.324

References:

[1]	M. Altwegg: Zur Axiomatik der teilweise geordneten Mengen. Comment. Math. Helv. 24 (1950), 149-155. · Zbl 0041.37704 · doi:10.1007/BF02567030
[2]	H.-J. Bandelt J. Hedlíková: Median algebras. Discrete Math. 45 (1983), 1 - 30. · Zbl 0506.06005 · doi:10.1016/0012-365X(83)90173-5
[3]	H. Draškovičová: Über die Relation ”zwischen” in Verbänden. Mat. Fyz. Čas. SAV 16 (1966), 13-20. · Zbl 0154.00901
[4]	D. W. Dubois: On partly ordered fields. Proc. Amer. Math. Soc. 7 (1956), 918 - 930. · Zbl 0071.26304 · doi:10.2307/2033563
[5]	J. Hashimoto: A ternary operation in lattices. Math. Japon. 2 (1951), 49-52. · Zbl 0044.02102
[6]	J. R. Isbell: Median algebra. Trans. Amer. Math. Soc. 260 (1980), 319 - 362. · Zbl 0446.06007 · doi:10.2307/1998007
[7]	L. M. Kelly: The geometry of normed lattices. Duke Math. J. 19 (1952), 661 - 669. · Zbl 0048.02401 · doi:10.1215/S0012-7094-52-01970-4
[8]	S. A. Kiss: Semilattices and a ternary operation in modular lattices. Bull. Amer. Math. Soc. 54 (1948), 1176-1179. · Zbl 0034.16602 · doi:10.1090/S0002-9904-1948-09145-5
[9]	M. Kolibiar: Linien in Verbänden. Anal. Stiint. Univ. Iasi 1 (1965), 89-98. · Zbl 0151.01603
[10]	M. Kolibiar T. Marcisová: On a question of J. Hashimoto. Mat. Čas. 24 (1974), 179-185. · Zbl 0285.06008
[11]	K. Menger: Untersuchungen über allgemeine Metrik. Math. Annalen 100 (1928), 75-163. · JFM 54.0622.02 · doi:10.1007/BF01448840
[12]	Z. Piesyk: On the betweenness relation in a set of cardinality \(\neq 4\). Bull. Acad. Polon. Sci. Math. Astr. Phys. 25 (1977), 667-670. · Zbl 0371.06001
[13]	E. Pitcher M. F. Smiley: Transitivities of betweenness. Trans. Amer. Math. Soc. 52 (1942), 95-114. · Zbl 0060.06408 · doi:10.2307/1990155
[14]	W. Prenowitz: Partially ordered fields and geometries. Amer. Math. Monthly 53 (1946), 439-449. · Zbl 0060.32403 · doi:10.2307/2306242
[15]	W. Prenowitz: A contemporary approach to classical geometry. Amer. Math. Monthly 68 (1961), no. 1, Part II. · Zbl 0094.15402 · doi:10.2307/2311904
[16]	W. Prenowitz J. Jantosciak: Geometries and join spaces. J. Reine Angew. Math. 257 (1972), 100-128. · Zbl 0264.50002
[17]	M. F. Smiley: A comparison of algebraic, metric, and lattice betweenness. Bull. Amer. Math. Soc. 49 (1943), 246-252. · Zbl 0060.06406 · doi:10.1090/S0002-9904-1943-07888-3
[18]	W. Szmielew: Oriented and nonoriented linear orders. Bull. Acad. Polon. Math. Astr. Phys. 25 (1977), 659-665. · Zbl 0358.06001