A repeated bijection in an isotopism of quasigroups is called a companion of the third component. The last is called a pseudoisomorphism with the companion. Isotopy coincides with pseudoisomorphy∗ in the class of inverse property loops and with isomorphy in the class of commutative inverse property loops. This result is a generalization of the corresponding theorem for commutative Moufang loops...

Two constructions due to Drápal produce a group by modifying exactly one quarter of the Cayley table of another group. We present these constructions in a compact way, and generalize them to Moufang loops, using loop extensions. Both constructions preserve associators, the associator subloop, and the nucleus. We conjecture that two Moufang 2-loops of finite order n with equivalent associator ca...

By left magma-$e$-magma, I mean a set containingthe fixed element $e$, and equipped by two binary operations "$cdot$", $odot$ with the property $eodot (xcdot y)=eodot(xodot y)$, namelyleft $e$-join law. So, $(X,cdot,e,odot)$ is a left magma-$e$-magmaif and only if $(X,cdot)$, $(X,odot)$ are magmas (groupoids), $ein X$ and the left $e$-join law holds.Right (and two-sided) magma-$e$-magmas are de...

Certain two constructions, due to Drápal, produce a group by modifying exactly one quarter of the Cayley table of another group. We present these constructions in a compact way, and generalize them to Moufang loops, using loop extensions. Both constructions preserve associators, the associator subloop, and the nucleus. We conjecture that two Moufang 2-loops of finite order n with equivalent ass...

Annotated translation of Parastrophic-orthogonal quasigroups, Acad. Nauk Moldav. SSR, Inst. Mat.s Vychisl. Tsentrom, Kishinev, 1983, prepared by A.D.Keedwell and P.Syrbu based on the original Russian and on an earlier English translation supplied to the rst author by Belousov himself. The notion of orthogonality plays an important role in the theory of Latin squares, and consequently also in th...

A biased expansion of a graph is a kind of branched covering graph with additional structure related to combinatorial homotopy of circles. Some but not all biased expansions are constructed from groups (group expansions); these include all biased expansions of complete graphs (assuming order at least four), which correspond to Dowling’s lattices of a group and encode an iterated group operation...

Hexagonal quasigroup is idempotent, medial and semisymmetric quasigroup. In this article we define and study vectors, sum of vectors and transfers. The main result is the theorem on isomorphism between the group of vectors, group of transfers and the Abelian group from the characterization theorem of the hexagonal quasigroups. 1. Hexagonal quasigroup Hexagonal quasigroups are defined in article...

We present new criteria for a multary (or polyadic) quasigroup to be isotopic to an iterated group operation. The criteria are consequences of a structural analysis of biased expansion graphs. We mention applications to transversal designs and generalized Dowling geometries. 1. Associativity in multary quasigroups A multary quasigroup is a set with an n-ary operation for some finite n ≥ 2, say ...

Let F be a family of finite loops closed under subloops and factor loops. Then every loop in F has the strong Lagrange property if and only if every simple loop in F has the weak Lagrange property. We exhibit several such families, and indicate how the Lagrange property enters into the problem of existence of finite simple loops. The two most important open problems in loop theory, namely the e...

Evolution algebras are not necessarily associative algebras satisfying eiej = 0 whenever ei, ej are two distinct basis elements. They mimic the self-reproduction of alleles in non-Mendelian genetics. We present elementary mathematical properties of evolution algebras that are of importance from the biological point of view. Several models of Mendelian [2, 4, 12, 6, 8, 11] and non-Mendelian gene...