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