Paramedial groupoids
Parastrophically Equivalent Quasigroup Equations
Parity of orthogonal automorphisms
Parity of orthogonal permutations
Permutable groupoids
Plenary stable quasigroups
Some results are obtained for quasigroups in which the operation of iterated squaring always leads to stability irrespective of the initial element.
Polyabelian loops and Boolean completeness
We consider the question of which loops are capable of expressing arbitrary Boolean functions through expressions of constants and variables. We call this property Boolean completeness. It is a generalization of functional completeness, and is intimately connected to the computational complexity of various questions about expressions, circuits, and equations defined over the loop. We say that a loop is polyabelian if it is an iterated affine quasidirect product of Abelian groups; polyabelianness...
Possible orders of nonassociative Moufang loops
The paper surveys the known results concerning the question: “For what values of does there exist a nonassociative Moufang loop of order ?” Proofs of the newest results for odd, and a complete resolution of the case even are also presented.
Powers and alternative laws
A groupoid is alternative if it satisfies the alternative laws and . These laws induce four partial maps on
Powers of elements in Jordan loops
A Jordan loop is a commutative loop satisfying the Jordan identity . We establish several identities involving powers in Jordan loops and show that there is no nonassociative Jordan loop of order .
Products of all elements in a loop and a framework for non-associative analogues of the Hall-Paige conjecture.
Products of partially ordered quasigroups
We describe necessary and sufficient conditions for a direct product and a lexicographic product of partially ordered quasigroups to be a positive quasigroup. Analogous questions for Riesz quasigroups are studied.
Pseudoautomorphisms of Bruck loops and their generalizations
We show that in a weak commutative inverse property loop, such as a Bruck loop, if is a right [left] pseudoautomorphism with companion , then [] must lie in the left nucleus. In particular, for any such loop with trivial left nucleus, every right pseudoautomorphism is an automorphism and if the squaring map is a permutation, then every left pseudoautomorphism is an automorphism as well. We also show that every pseudoautomorphism of a commutative inverse property loop is an automorphism, generalizing...
Quadratic Level Quasigroup Equations With Four Variables II: the Lattice of Varieties
Quantum idempotence, distributivity, and the Yang-Baxter equation
Quantum quasigroups and loops are self-dual objects that provide a general framework for the nonassociative extension of quantum group techniques. They also have one-sided analogues, which are not self-dual. In this paper, natural quantum versions of idempotence and distributivity are specified for these and related structures. Quantum distributive structures furnish solutions to the quantum Yang-Baxter equation.
Quasigroup automorphisms and symmetric group characters
The automorphisms of a quasigroup or Latin square are permutations of the set of entries of the square, and thus belong to conjugacy classes in symmetric groups. These conjugacy classes may be recognized as being annihilated by symmetric group class functions that belong to a -ideal of the special -ring of symmetric group class functions.
Quasigroup covers of division groupoids
Let be a division groupoid that is not a quasigroup. For each regular cardinal we construct a quasigroup on that is a quasigroup cover of (i.e., is a homomorphic image of and is not an image of any quasigroup that is a proper factor of ). We also show how to easily obtain quasigroup covers from free quasigroups.
Quasigroups and Factorisation of Complete Digraphs
Quasigroups and tactical systems.
Quasigroups arisen by right nuclear extension
The aim of this paper is to prove that a quasigroup with right unit is isomorphic to an -extension of a right nuclear normal subgroup by the factor quasigroup if and only if there exists a normalized left transversal to in such that the right translations by elements of commute with all right translations by elements of the subgroup . Moreover, a loop is isomorphic to an -extension of a right nuclear normal subgroup by a loop if and only if is middle-nuclear, and there exists...