The Bernstein problem for type \((n-2,2)\)
type: Item
instance of: scholarly article
title: The Bernstein problem for type \((n-2,2)\)
zbMATH Open document ID: 0859.17019
author: Juan C. Gutierrez Fernandez
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: A full description of stochastic nonregular nonexceptional nonnuclear Bernstein algebras of type \((n-2,2)\) is obtained. So, if the Lyubich conjecture is proved to be true, the above result would completely solve the Bernstein problem for the type \((n-2,2)\). The first section is devoted to introducing the Bernstein problem, concerning mathematical expressions of fundamental laws of biological heredity, and the basic related concepts. We recall that a Bernstein algebra \((A,\omega)\) is a (commutative) real algebra \(A\) endowed with a nonzero algebra homomorphism \(\omega: A\to\mathbb{R}\) satisfying \((x^2)^2= \omega(x)^2x^2\) for any \(x\in A\). It is said that the Bernstein algebra \(A\) is a stochastic Bernstein algebra if it has a finite basis (named stochastic basis) such that the simplex \(\Delta\) spanned by this basis is invariant with respect to the multiplication \((\Delta\cdot \Delta\subset \Delta)\). Such an algebra has a nonzero idempotent \(e\) which gives a Peirce decomposition \(A=\text{Re}\oplus U_e\oplus V_e\), where \(\text{ker} (\omega)= U_e\oplus V_e\), \(U_e=\{x\in A:ex={1\over2}x\}\) and \(V_e=\{x\in A:ex=0\}\). It is a well-known fact that \(\dim U_e\) (and then \(\dim V_e\)), \(\dim U^2_e\) and \(\dim(u_eV_e+ V^2_e)\) do not depend on the choice of the idempotent element, so the type \(A=(m,\delta)\) can be defined where \(m-1=\dim U_e\) and \(\delta= \dim V_e\) (so \(n=\dim A=m+\delta)\). Following definitions introduced by Lyubich, if \(U_eV_e+ V^2_e=0\), the Bernstein algebra \(A\) is called regular, and if \(U^2_e=0\) is exceptional. Every Bernstein algebra of type \((m,\delta)\) with \(m-1\) or \(\delta\) less than or equal to 1 is regular or exceptional. An algebra \(A\) is called nuclear if \(A^2=A\). The Bernstein problem is reduced to describing all stochastic Bernstein algebras and the multiplication with respect to a stochastic basis. Several partial solutions to this problem have been obtained in the last years. The second se…
Mathematics Subject Classification ID: 17D92
zbMATH DE Number: 883778
zbMATH Keywords: Bernstein algebras
zbMATH Keywords: Bernstein problem
zbMATH Keywords: stochastic Bernstein algebra
zbMATH Keywords: stationary evolutionary operators
zbMATH Keywords: Lyubich conjecture
reviewed by: A. Castellón Serrano
MaRDI profile type: Publication
full work available at URL: jabr.1996.0138
OpenAlex ID: W2020107205
DOI: 10.1006/JABR.1996.0138
description: scientific article; zbMATH DE number 883778