Óscar J. Falcón

Classification of asexual diploid organisms by means of strongly isotopic evolution algebras defined over any field

Abstract Evolution algebras were introduced into Genetics to deal with the mechanism of inheritance of asexual organisms. Their distribution into isotopism classes is uniquely related with the mutation of alleles in non-Mendelian Genetics. This paper deals with such a distribution by means of Computational Algebraic Geometry. We focus in particular on the two-dimensional case, which is related to the asexual reproduction processes of diploid organisms. Specifically, we determine the existence of four isotopism classes, whatever the base field is, and we characterize the corresponding isomorphism classes.

Counting and enumerating partial Latin rectangles by means of computer algebra systems and CSP solvers

This paper provides an in-depth analysis of how computational algebraic geometry can be used to deal with the problem of counting and classifying r × s partial Latin rectangles based on n symbols of a given size, shape, type or structure. The computation of Hilbert functions and triangular systems of radical ideals enables us to solve this problem for all r, s, n ≤ 6. As a by-product, explicit formulas are determined for the number of partial Latin rectangles of size up to six. We focus then on the study of non-compressible regular partial Latin squares and their equivalent incidence structure called seminet, whose distribution into main classes is explicitly determined for point rank up to eight. We prove in particular the existence of two new configurations of point rank eight.

Algebraic computation of genetic patterns related to three-dimensional evolution algebras

The mitosis process of an eukaryotic cell can be represented by the structure constants of an evolution algebra. Any isotopism of the latter corresponds to a mutation of genotypes of the former. This paper uses Computational Algebraic Geometry to determine the distribution of three-dimensional evolution algebras over any field into isotopism classes and hence, to describe the spectrum of genetic patterns of three distinct genotypes during a mitosis process. Their distribution into isomorphism classes is also determined in case of dealing with algebras having a one-dimensional annihilator.

Computation of isotopisms of algebras over finite fields by means of graph invariants

In this paper we define a pair of faithful functors that map isomorphic and isotopic finite-dimensional algebras over finite fields to isomorphic graphs. These functors reduce the cost of computation that is usually required to determine whether two algebras are isomorphic. In order to illustrate their efficiency, we determine explicitly the classification of two- and three-dimensional partial quasigroup rings.

Isomorphism and isotopism classes of filiform Lie algebras of dimension up to seven

This paper deals with a new series of isotopism invariants that enable us to determine explicitly the distribution of n-dimensional filiform Lie algebras into isomorphism and isotopism classes. For

Isotopism and Isomorphism Classes of Certain Lie Algebras over Finite Fields

n\le 6

A computational algebraic geometry approach to enumerate Malcev magma algebras over finite fields: Ó. J. FALCÓN, R. M. FALCÓN AND J. NÚÑEZ

n≤6, this distribution is explicitly obtained over any field. For