Multi-algebra duplication.

*(English)*Zbl 0658.92015The aim of this paper is to use multi-algebra duplication to simplify the proofs of J. Lopez-Sanchez, and A. Perez de Vargas [Bull. Math. Biol. 47, 771-782 (1985; Zbl 0586.92016)], and to show that this idea can be usefully applied to other situations. The author starts, however, with a brief, but very lucid introduction to some of the basic algebraic ideas in genetics - this should prove very useful to readers not very familiar with this subject. For a full account of algebra in genetics he rightly refers the reader to A. Wörz-Busekros [Algebras in genetics. Lect. Notes in Biomathematics 36 (1980; Zbl 0431.92017)].

Multi-algebra duplication of two algebras having a common basis is defined in terms of the duplication of either algebra [see the author’s paper, Proc. Edinburgh Math. Soc., II. Ser. 17, 289-298 (1971; Zbl 0247.92002)], and his two main theorems are concerned with subalgebras of such a duplication, and the train roots. [In Theorem 1, the first lower case “c” should be replaced by an upper case “C”.]

Multi-algebra duplication of two algebras having a common basis is defined in terms of the duplication of either algebra [see the author’s paper, Proc. Edinburgh Math. Soc., II. Ser. 17, 289-298 (1971; Zbl 0247.92002)], and his two main theorems are concerned with subalgebras of such a duplication, and the train roots. [In Theorem 1, the first lower case “c” should be replaced by an upper case “C”.]

Reviewer: E.W.Wallace

##### Keywords:

recombination rates; linked loci; factoring; multi-algebra duplication; subalgebras; train roots
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##### References:

[1] | Gonshor, H.: Special train algebras arising in genetics. Proc. Edinb. Math. Soc. 12, 41-53 (1980) · Zbl 0249.17003 |

[2] | Gonshor, H.: Special train algebras arising in genetics II. Proc. Edinb. Math. Soc. 14, 333-338 (1965) · Zbl 0139.03102 |

[3] | Gonshor, H.: Contributions to genetic algebras. Proc. Edinb. Math. Soc. 17, 289-298 (1977) · Zbl 0247.92002 |

[4] | Gonshor, H.: Contributions to genetic algebras II. Proc. Edinb. Math. Soc. 18, 273-279 (1973) · Zbl 0272.92012 |

[5] | Lopez-Sanchez, J., Perez de Vargas, A.: Zygotic algebra for two linked loci with sexually different recombination rates. Bull. Math. Biol. 47, 771-782 (1985) · Zbl 0586.92016 |

[6] | Worz-Busekros, A.: Algebras in genetics. Lect. Notes Biomath. 36. Berlin Heidelberg New York: Springer 1980 · Zbl 0431.92017 |

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