# zbMATH — the first resource for mathematics

Planar ternary rings with zero of translation, Moufang and Desarguesian planes. (English) Zbl 0569.51002
In this paper the algebraic properties of planar ternary rings with zero (PTRZ’s) of translation, Moufang and Desarguesian planes are given. The main theorems are as follows: Let $$(S,<>)$$ be a coordinatizing PTRZ of a projective plane $$\pi$$ relative to the reference points X, Y, 0.
Theorem 1. $$\pi$$ is a translation plane with respect to XY if and only if $$(S,<>)$$ satisfies $$A)\quad <a,m,b>=a.m+b.$$ B) $$(S,+)$$ is a group. C) $$c\in S$$ determined by $$am+bm=cm$$ for a,b$$\in S$$, $$m\in S^*$$ is independent of m.
Theorem $$2: \pi$$ is a Moufang plane if and only if $$(S,<>)$$ satisfies A), B), C) and D) $$c\in S$$ determined by $$ma+mb=mc$$ for a,b$$\in S$$, $$m\in S^*$$ is independent of m. Theorem 3. $$\pi$$ is a Desarguesian plane if and only if $$(S,<.>)$$ satisfies A), B), C), D) and F) $$c\in S$$ determined by $$am=c(d\setminus bm)$$ or $$a,b,d,m\in S^*$$ is independent of m. (For $$a=0$$; $$b,d,m\in S^*$$ the condition is trivially satisfied with $$c=0.)$$
##### MSC:
 51A25 Algebraization in linear incidence geometry 51A30 Desarguesian and Pappian geometries 12K99 Generalizations of fields 20N10 Ternary systems (heaps, semiheaps, heapoids, etc.)
Full Text:
##### References:
 [1] M. Bommireddy, L. N. Reddy andY. B. Reddy, Remarks on Planar Ternary Rings coordinatizing a projective plane. Arch. Math.42, 573-576 (1984). · Zbl 0537.51004 [2] M. Bommireddy andL. Nagamuni Reddy, Comparison and inversion of planar ternary rings with zero. Arch. Math.43, 562-565 (1984). · Zbl 0534.16033 [3] M.Hall, The theory of Groups. New York 1959. · Zbl 0084.02202 [4] D. Kluck? andL. Markov?, Ternary rings with zero associated to translation planes. Czech. Math. J.23 (98), 617-628 (1973). · Zbl 0273.50017 [5] D. Kluck?, Ternary rings with zero associated to Desarguesian and Pappian planes. Czech. Math. J.24 (99), 607-613 (1974). · Zbl 0329.50016 [6] G.Pickert, Projektive Ebenen. Berlin 1975. [7] L. A. Skornyakov, Natural domains of Veblen-Wedderburn projective planes. Russ. Izv. Nauk. SSSR. Ser. Mat.13, 447-472 (1949), Amer. Math. Soc. Transl.58 (1951). · Zbl 0033.12502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.