Raúl Falcón | Searcharxiv

arXiv2026

A historical perspective of Tian's evolution algebras

Even if it has been less than a decade and a half since Tian introduced his concept of evolution algebras to represent algebraically non-Mendelian rules in Genetics, their study is becoming increasingly widespread mainly due to their applications to many scientific disciplines. In order to facilitate further research on the topic, this paper deals with the past and present research on these kind of algebras, together with the most relevant topics regarding them.

math.HO↗

arXiv2023

Latin bitrades derived from quasigroup autoparatopisms

In 2008, Cavenagh and Drápal, et al, described a method of constructing Latin trades using groups. The Latin trades that arise from this construction are entry-transitive (that is, there always exists an autoparatopism of the Latin trade mapping any ordered triple to any other ordered triple). Moreover, useful properties of the Latin trade can be established using properties of the group. However, the construction does not give a direct embedding of the Latin trade into any particular Latin square. In this paper, we generalize the above to construct Latin trades embedded in a Latin square $L$, via the autoparatopism group of the quasigroup with Cayley table $L$. We apply this theory to identify non-trivial entry-transitive trades in some group operation tables as well as in Latin squares that arise from quadratic orthomorphisms

math.CO↗