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Genetic algebras associated with polyploidy. (English) Zbl 0144.27202
This paper continues the studies, initiated by I. M. H. Etherington, of certain types of nonassociative algebras which have a bearing on genetic theories. H. Gonshor [Proc. Edinb. Math. Soc., II. Ser. 12, 41–53 (1960; Zbl 0249.17003); ibid. 14, 333–338 (1965; Zbl 0139.03102)] has recently considered algebras corresponding to polyploidy. In the present paper the author defines a segregation algebra to be one with a basis \(D_a\) \((a = 0,\dots, n)\) and multiplication given by
\[ D_aD_b = \binom{2sn}{n}^{-1} \sum_{i=0}^n \binom{sa+sb}{i}\binom{2sn-sa-sb}{n-i} D_i. \]
The cases \(s=1\) and \(2\) correspond to chromosome and chromatid segregation, respectively. Certain natural types of inheritance give rise to mixtures of algebras of this type. Algebras corresponding to a fixed ploidy are isotopic in a special way, but there are algebras which occur in certain genetic systems and which are not isotopic to segregation algebras.

MSC:
17D92 Genetic algebras
92D10 Genetics and epigenetics
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