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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

Orin Chein, Andrew Rajah (2000)

Commentationes Mathematicae Universitatis Carolinae

The paper surveys the known results concerning the question: “For what values of $n$ does there exist a nonassociative Moufang loop of order $n$ ?” Proofs of the newest results for $n$ odd, and a complete resolution of the case $n$ even are also presented.

Powers and alternative laws

Nicholas Ormes, Petr Vojtěchovský (2007)

Commentationes Mathematicae Universitatis Carolinae

A groupoid is alternative if it satisfies the alternative laws $x (x y) = (x x) y$ and $x (y y) = (x y) y$ . These laws induce four partial maps on $ℕ^{+} \times ℕ^{+}$ $(r, s) \mapsto ...$

Powers of elements in Jordan loops

Kyle Pula (2008) Commentationes Mathematicae Universitatis Carolinae

A Jordan loop is a commutative loop satisfying the Jordan identity

(x^{2} y) x = x^{2} (y x)

. We establish several identities involving powers in Jordan loops and show that there is no nonassociative Jordan loop of order

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Products of all elements in a loop and a framework for non-associative analogues of the Hall-Paige conjecture.

Pula, Kyle (2009)

The Electronic Journal of Combinatorics [electronic only]

Products of partially ordered quasigroups

Milan Demko (2008)

Commentationes Mathematicae Universitatis Carolinae

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

Mark Greer, Michael Kinyon (2012)

Commentationes Mathematicae Universitatis Carolinae

We show that in a weak commutative inverse property loop, such as a Bruck loop, if $α$ is a right [left] pseudoautomorphism with companion $c$ , then $c$ [ $c^{2}$ ] 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...

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