arXiv subjects
Publications and source records attributed to Andrés Pérez-Rodríguez.
In this paper, we study modularity in the context of evolution algebras. Although this property has been previously considered, a complete description is still missing in several natural settings. In particular, we obtain a full characterisation of modular evolution algebras in the nilpotent case and in the class of supersolvable regular evolution algebras.
This short note provides positive answers to two conjectures of Camacho, Khudoyberdiyev, and Omirov on the classification of complete evolution algebras. Our approach is based on analysing the solution set of a generic non-linear polynomial system of equations using elementary tools from algebraic geometry. We also obtain new results on subalgebras and idempotents of evolution algebras, and conclude by proposing a conjecture that may characterise solvable evolution algebras.
The main purpose of this paper is to study formal deformations of evolution algebras, determining their existence and classifying them up to equivalence. In addition, we examine degenerations in this setting and provide Hasse diagrams that describe the degeneration relations among nilpotent evolution algebras of dimensions up to four.
This paper develops a Frattini theory for evolution algebras defining the Frattini subalgebra as the intersection of all maximal subalgebras, and the Frattini ideal as the largest ideal contained in it. To this end, we revisit the notion of nilradical, whose classical definition is not directly applicable in this setting, and propose the supersolvable nilradical as a suitable alternative. This leads to necessary and sufficient conditions for the triviality of the Frattini subalgebra and ideal. Finally, we also briefly examine the relevance of the Frattini ideal in the study of dually atomistic evolution algebras.
The main goal of this note is to show that subalgebras of regular evolution algebras are themselves evolution algebras. This allows us to assume, without loss of generality, that every subalgebra in the regular setting has a basis consisting of vectors with disjoint supports. Finally, we use this result to characterise the existence of codimension-one subalgebras in regular evolution algebras.
This article is devoted to studying gonosomal algebras and operators with a single male genotype. We compute the limit points of the trajectories of the corresponding normalised gonosomal operators, describing the development of specific populations and providing the corresponding biological interpretations.
The main objective of this paper is to study the relationship between a solvable evolution algebra and its subalgebra lattice, emphasizing two of its main properties: distributivity and modularity. First, we will focus on the nilpotent case, where distributivity is characterised, and a necessary condition for modularity is deduced. Subsequently, we comment on some results for solvable non-nilpotent evolution algebras, finding that the ones with maximum index of solvability have the best properties. Finally, we characterise modularity in this particular case by introducing supersolvable evolution algebras and computing the terms of the derived series.