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 nonMendelian Genetics. This paper deals with such a distribution by means of Computational Algebraic Geometry. We focus in particular on the twodimensional 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.
This paper provides an in‐depth analysis of how computer algebra systems and CSP solvers can be used to deal with the problem of enumerating and distributing the set of r×s partial Latin rectangles based on n symbols according to their weight, 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 weight up to 6. Further, to illustrate the effectiveness of the computational method, we focus on the enumeration of 3 subsets: (1) noncompressible and regular, (2) totally symmetric, and (3) totally conjugate orthogonal partial Latin squares. In particular, the former enables us to enumerate the set of seminets of point rank up to 8 and to prove the existence of 2 new configurations of point rank 8. Finally, as an illustrative application, it is also exposed a method to construct totally symmetric partial Latin squares that gives rise, under certain conditions, to new families of Lie partial quasigroup rings.
A computational algebraic geometry approach to enumerate Malcev magma algebras over finite fields
Communicated by J. Vigo-AguiarThe set M n .K/ of n-dimensional Malcev magma algebras over a finite field K can be identified with algebraic sets defined by zero-dimensional radical ideals for which the computation of their reduced Gröbner bases makes feasible their enumeration and distribution into isomorphism and isotopism classes. Based on this computation and the classification of Lie algebras over finite fields given by De Graaf and Strade, we determine the mentioned distribution for Malcev magma algebras of dimension n Ä 4. We also prove that every three-dimensional Malcev algebra is isotopic to a Lie magma algebra. For n D 4, this assertion only holds when the characteristic of the base field K is distinct of two.In this section, we expose some basic concepts and results on isotopisms of algebras, Malcev algebras, and computational algebraic geometry that are used throughout the paper. For more details about these topics, we refer to the original articles of Albert [16], Malcev [3], and Sagle [4] and to the monographs of Cox, Little, and O'Shea [39, 40].2.
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 threedimensional partial quasigroup rings.
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 onedimensional annihilator.
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