Genetic Algebras Research Articles

On train algebras of rank 4

The existence of idempotent elements in train algebras of rank greater than 3 is an open question to be solved. Recent H. Guzzo results [7] on train algebras of rank 4 are based on the underlying assumption of the existence of an idempotent. In the present paper we establish the conditions that ensure the existence of such an idempotent. We also give additional properties on the Peirce decomposition which allow us to characterize some train algebras of rank 4. Finally, we give a characterization of the train algebras of rank 4 which are power-associative algebras or Jordan algebras.

Shape identities in genetic algebras

We introduce the concepts of shape identity and level for nonassociative algebras and establish some properties in the setting of genetic algebras.

Embedding nil algebras in train algebras

We generalize the classical example, due to Abraham, of a train algebra that is not special train, to non necessarily commutative right nil algebras of index n.

Long-time behavior of continuous time models in genetic algebras.

In [2] the solutions of Andreoli's differential equation in genetic algebras with genetic realization were shown to converge to equilibria. Here we derive an explicit formula for these limits.

Indecomposable baric algebras. II

We exhibit some classes of baric algebras over a field for which the Krull-Schmidt theorem, proved in the previous paper “Indecomposable baric algebras,” is valid. We prove also that the commutative duplicate of an indecomposable commutative baric algebra A such that A 2 = A is indecomposable. Two more examples of indecomposable baric algebras, relevant in genetics, are given: linked and independent loci.

E-ideals in baric algebras

E-ideals in baric algebras

The peirce decomposition for commutative train algebras

In this paper we obtain a Peirce decomposition for some train algebras of rank n; the multiplication table for the proper subspaces is given. Some additional results for train algebras of rank 4 are proved and a description of those train algebras of rank 4, having A ½ = 0, is given by pairs of bilinear mappings.

A note on Baric algebras

In this paper we present a characterization of baric algebras. In particular we study those in which the identity x3 = w(x)x2 holds. Moreover, for every field K, we prove that this identity guarantees that the annihilator of Ker (w) is an ideal in A and we give example of a subspace of Ker (w) whose annihilator is not an ideal.

Les pol( n,m)-algèbres: identités polynômiales symétriques dans des algèbres

Symmetric polynomial identities arise in several algebras. In this work we study nonassociative, noncommutative algebras verifying these kinds of identities. The equivalence between these algebras and algebras satisfying particular monomial relations (in which a linear form appears) is proved. We thus present simplified definitions of some classical baric algebras, for instance, Bernstein's algebras.

Indecomposable baric algebras

We introduce the concept of indecomposable baric algebras and prove a Krull-Schmidt theorem for algebras satisfying ascending and descending chain conditions. We analyze some well-known examples of generic algebras with regard to their indecomposability. Counterexamples are given to three natural statements about indecomposability.

Associative bilinear forms in some baric algebras

Let ( A, ω) be a K-algebra, e ϵ A a nontrivial idempotent, K an infinite field whose characteristic is different from 2, and A = K e ⊕ U ⊕ V, where U = { x ϵ ker ω| e x = 1 2 x}. If B : A × A → K is an associative bilinear form, we prove the two following main facts: (1) If A is a kth order Bernstein algebra and B is a nondegenerate form, then A is a kth order quasiconstant algebra and the idempotent is unique. (2) If x 3 - (1 + γ)ω( x) x 2 + γω( x) 2 x = 0 is the rank equation of A and if B is nondegenerate, then A is a Jordan algebra. If 0 ≠ γ ≠ 1, then B , the symmetry of B only depends on the symmetry of B | v , and B ( e 1, e 1) = B ( e, e) for any idempotent elements e, e 1 in A.

On continuous time models in genetic and Bernstein algebras

We discuss the long-time behavior of Andreoli's differential equation for genetic algebras and for Bernstein algebras and show convergence to an equilibrium in both cases. For a class of Bernstein algebras this equilibrium is determined explicitly.

The multilocus Hardy-Weinberg law

Work on the genetic algebra of multilocus genetic systems is reviewed, with particular emphasis on aspects of Hardy-Weinberg theory, including existence and stability of equilibria, global convergence and disequilibrium functions. It is pointed out that the non-uniqueness of the disequilibrium functions does not necessarily invalidate proof of the multilocus Hardy-Weinberg law.

The Frattini subalgebra of a Bernstein algebra

Let A be a finite-dimensional Bernstein algebra over a field K with characteristic not 2. Maximal subalgebras of A are studied, and they are determined if A is a genetic algebra. It is also proved that the intersection of all maximal subalgebras of A (the Frattini subalgebra of A) is always an ideal. Finally the structure of Bernstein algebras with Frattini subalgebra equal to zero is described.

Biometric and chromosome algebras

This note continues the development of the infinite-dimensional genetic algebra approach to problems of population genetics. Two algebras are studied. One describes the familiar problem of a quantitative characteristic, and the other provides a way of treating the whole chromosome as an entity.

Biometric and chromosome algebras

This note continues the development of the infinite-dimensional genetic algebra approach to problems of population genetics. Two algebras are studied. One describes the familiar problem of a quantitative characteristic, and the other provides a way of treating the whole chromosome as an entity.

On Bernstein algebras which are train algebras

The class of non-associative algebras over a field of characteristic zero named in the title is studied using a result of Ouattara [9]. As an application, the differential equation for overlapping generations in the time-continuous model is solved.

Sur les algèbres de bernstein qui sont des T-algèbres

In this note we show that for finite-dimensional Bernstein algebras over a field of characteristic different from 2: 1. (1) The principal nilpotency of an element implies its strong nilpotency. 2. (2) If Ker(ω) is a nil algebra, then it is a train algebra. 3. (3) Every train algebra of dimension n ⩽ 5 is special train algebra, and that is a best possible result.

On train algebras of rank 3

We classify train algebras of rank 3, having type ( n, 1) or (3, n − 2), for an arbitrary value of n. Then we classify train algebras of rank 3 which have an anisotropic kernel.

Sur les T-algèbres de Jordan

We give here some results about train algebras which are Jordan. In particular we show that for train algebras of rank 3, being Jordan is equivalent to being power associative. This is not true for train algebras of rank > 3.