On classification of associative non-division genetic algebras

• Main Author: Ganikhodjaev, Nasir
• Format: Conference or Workshop Item
• Language: English
• Published: 2013
• Subjects: QA Mathematics
• Online Access: http://irep.iium.edu.my/28943/1/Prof_Nasir--ICMSS2013.pdf; http://irep.iium.edu.my/28943/

General genetic algebras are the product of interaction between biology and mathematics. The study of these algebras reveals the algebraic structure of Mendelian and non-Mendelian genetics, which always simplifies and shortens the way to understand the genetic and evolutionary phenomena in real world. Mathematically, the algebras that arise in genetics are very interesting structures. Many of the algebraic properties of these structures have genetic significance. Indeed, the interplay between the purely mathematical structure and the corresponding genetic properties makes this subject so fascinating. Let a quadratic stochastic operator V: S"- 1 ~S"- 1 be a genetic realization, where V is defined by cubic matrix {pij,k : i,j,k= l, ... ,n} such that a) PiJ,k ?:. 0; b) PiJ,k= Pji,k and c) L P!i,k = 1 . k=l An algebra R with genetic realization V is an real algebra which has a basis {a1, a2, .. . ,an} and a multiplication table ai;aj = !i,kak k=l Here PiJ.k is a frequency that the next generation reproduced by two gametes carrying ai and a1 will inherit ak, k=l, .. . ,n. An associative algebra R is a division algebra if it has a multiplicative identity element e =/= 0 and every non-zero element has a multiplicative inverse. In general, the algebras which arise in genetics are non-division algebra. In this paper we describe genetic significance of non-invertible elements, present a construction of non-division genetic algebras, and investigate the problem of their classification.