New closure conditions and some problems in loop theory.
Non-associative geometry and discrete structure of spacetime
A new mathematical theory, non-associative geometry, providing a unified algebraic description of continuous and discrete spacetime, is introduced.
Nonassociative triples in involutory loops and in loops of small order
A loop of order possesses at least associative triples. However, no loop of order that achieves this bound seems to be known. If the loop is involutory, then it possesses at least associative triples. Involutory loops with associative triples can be obtained by prolongation of certain maximally nonassociative quasigroups whenever is a prime greater than or equal to or , an odd prime. For orders the minimum number of associative triples is reported for both general and involutory...
Nonsplitting F-quasigroups
T. Kepka, M.K. Kinyon and J.D. Phillips: The structure of F-quasigroups, J. Algebra 317 (2007), no. 2, 435–461 developed a connection between F-quasigroups and NK-loops. Since NK-loops are contained in the variety generated by groups and commutative Moufang loops, a question that arises is whether or not there exists a nonsplit NK-loop and likewise a nonsplit F-quasigroup. Here we prove that there do indeed exist nonsplit F-quasigroups and show that there are exactly four corresponding nonsplit...
Normal subsets of quasigroups
Normality, nuclear squares and Osborn identities
Let be a loop. If is such that for each standard generator of Inn, then does not have to be a normal subloop. In an LC loop the left and middle nucleus coincide and form a normal subloop. The identities of Osborn loops are obtained by applying the idea of nuclear identification, and various connections of Osborn loops to Moufang and CC loops are discussed. Every Osborn loop possesses a normal nucleus, and this nucleus coincides with the left, the right and the middle nucleus. Loops that...
Not every Osborn loop is universal.
Note on analytic Moufang loops
It is explicitly shown how the Lie algebras can be associated with the analytic Moufang loops. The resulting Lie algebra commutation relations are well known from the theory of alternative algebras.
Notes on associative triples of elements in commutative groupoids
Notes on distributive groupoids
Notes on left distributive groupoids
Notes on quasimodules
Notes on the number of associative triples
n-арные инверсные квазигруппы (J-квазигруппы)
Odd order semidirect extensions of commutative automorphic loops
We analyze semidirect extensions of middle nuclei of commutative automorphic loops. We find a less complicated conditions for the semidirect construction when the middle nucleus is an odd order abelian group. We then use the description to study extensions of orders and .
On (2,2) modular law for ternary GD-groupoids
On -basic quasigroups and their congruences
On a class of distributive Steiner quasigroups
On a class of Moufang loops
On a Class of N-ary Quasigroups