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Comparison and inversion of planar ternary rings with zero. (English) Zbl 0534.16033
In this paper the properties of comparative and inverse of a planar ternary ring with zero (PTRZ) are studied. Let \((S,< >)\) be a PTRZ and \(s\in S\). Then \((S,< >_ s)\) is called the comparative of \((S,< >)\) with respect to s if \(<a,s+m,b>=<a,s,<a,m,b>_ s>\) and \((S,< >^ i)\) is called the inverse of \((S,< >)\) if \(m\neq 0,\quad<b,m,d>=0\Rightarrow<<a,m,b>^ i,m,d>=a.\) It is proved that (i) the comparative (inverse) of a PTRZ is a PTRZ, (ii) the comparative of a generalized Cartesian group is a generalized Cartesian group, (iii) the projective planes induced by a PTRZ and its comparative (inverse) are isomorphic \((iv)\quad< >=< >_{s,-s}\) and \((v)\quad< >=< >^{i,i}.\)

MSC:
16Y60 Semirings
17A40 Ternary compositions
51E15 Finite affine and projective planes (geometric aspects)
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[1] G.Pickerty, Projektive Ebenen. Berlin 1975.
[2] L. A. Skornyakov, Natural domains of Veblen-Wedderburn projective planes. Russ. Izv. Nauk, SSSR., Ser. Mat.13, 447-472 (1949); Amer. Math. Soc.58 (1951). · Zbl 0033.12502
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