Schreier loops
We study systematically the natural generalization of Schreier's extension theory to obtain proper loops and show that this construction gives a rich family of examples of loops in all traditional common, important loop classes.
Secure two-way on-line communication by using quasigroup enciphering with almost public key.
Selfdistributive groupoids of small orders
After enumerating isomorphism types of at most five-element left distributive groupoids, we prove that a distributive groupoid with less than 81 elements is necessarily medial.
Selfdistributive groupoids. Part A1. Non-indempotent left distributive groupoids
Selfdistributive groupoids. Part D1: Left distributive semigroups
Selfdistributive grupoids, Part A2: Non-idempotent left distributive quasigroups
Self-invariant 1-factorizations of complete graphs and finite Bol loops of exponent 2.
Self-orthogonal cyclic n-quasigroups.
Self-orthogonal cyclic n-quasigroups. (Short Communication).
Semigroup representations of medial groupoids
Semisymmetrization and Mendelsohn quasigroups
The semisymmetrization of an arbitrary quasigroup builds a semisymmetric quasigroup structure on the cube of the underlying set of the quasigroup. It serves to reduce homotopies to homomorphisms. An alternative semisymmetrization on the square of the underlying set was recently introduced by A. Krapež and Z. Petrić. Their construction in fact yields a Mendelsohn quasigroup, which is idempotent as well as semisymmetric. We describe it as the Mendelsohnization of the original quasigroup. For quasigroups...
Separable Quasigroups.
Simple paramedial groupoids
Simple quasigroups whose inner permutations commute
Simple quasigroups with commuting inner permutations are medial.
Skeletons in multigraphs
Under a multigraph it is meant in this paper a general incidence structure with finitely many points and blocks such that there are at least two blocks through any point and also at least two points on any block. Using submultigraphs with saturated points there are defined generating point sets, point bases and point skeletons. The main result is that the complement to any basis (skeleton) is a skeleton (basis).
Skewsquares in quadratical quasigroups
The concept of pseudosquare in a general quadratical quasigroup is introduced and connections to some other geometrical concepts are studied. The geometrical presentations of some proved statements are given in the quadratical quasigroup .
Smooth quasigroups and loops: forty-five years of incredible growth
The remarkable development of the theory of smooth quasigroups is surveyed.
Sobre un teorema de Lagrange para quase grupos de Ward
Sofic groups are not locally embeddable into finite Moufang loops
We shall show that there exist sofic groups which are not locally embeddable into finite Moufang loops. These groups serve as counterexamples to a problem and two conjectures formulated in the paper by M. Vodička, P. Zlatoš (2019).
Solution of Belousov's problem
The authors prove that a local n-quasigroup defined by the equation , where , i,j = 1,...,n, are arbitrary functions, is irreducible if and only if any two functions and , i ≠ j, are not both linear homogeneous, or these functions are linear homogeneous but . This gives a solution of Belousov’s problem to construct examples of irreducible n-quasigroups for any n ≥ 3.