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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.)
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