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Coordinatization of projective planes by special planar ternary rings. (English) Zbl 0961.51003

The notion of an admissible planar ternary ring (APTR) is introduced by a slight modification of the well-known planar ternary rings found e.g. in [D. R. Hughes and F. C. Piper, ‘Projective planes’ (1973; Zbl 0267.50018)]. A number of existing theorems concerning geometrical properties (transitivity) of the plane and algebraic properties of a coordinatizing APTR are reformulated and reproved.

MSC:

51C05 Ring geometry (Hjelmslev, Barbilian, etc.)
51A25 Algebraization in linear incidence geometry
20N10 Ternary systems (heaps, semiheaps, heapoids, etc.)

Citations:

Zbl 0267.50018
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References:

[1] Martin G. E.: Projective planes and isotopic ternary rings. Amer. Math. Monthly 74 (1967), 1185-1195. · Zbl 0164.51401
[2] Martin G. E.: Projective planes and isogeic ternary rings. Estratto ”Le matematiche”, 23, 1 (1968), 185-196. · Zbl 0188.24404
[3] Klucky D.: Natural planar ternary rings. Acta Univ. Palacki. Olomuc., Fac. rer. nat., 69 (1981), 47-57. · Zbl 0483.51003
[4] Klucky D., Markova L.: Ternary rings with a left quasi-zero belonging to translation planes. Acta Univ. Palacki. Olomuc., Fac. rer. nat., 65 (1980), 89-100. · Zbl 0492.51002
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