Corrigendum to “Realizations of Loops and Groups defined by short identities”
Realizations of Loops and Groups defined by short identities
In a recent paper, those quasigroup identities involving at most three variables and of “length” six which force the quasigroup to be a loop or group have been enumerated by computer. We separate these identities into subsets according to what classes of loops they define and also provide humanly-comprehensible proofs for most of the computer-generated results.
When is it hard to show that a quasigroup is a loop?
We contrast the simple proof that a quasigroup which satisfies the Moufang identity is necessarily a loop (Moufang loop) with the remarkably involved prof that a quasigroup which satisfies the Moufang identity is likewise necessarily a Moufang loop and attempt to explain why the proofs are so different in complexity.
Construction, properties and applications of finite neofields
We give a short account of the construction and properties of left neofields. Most useful in practice seem to be neofields based on the cyclic group and particularly those having an additional divisibility property, called . We shall give examples of applications to the construction of orthogonal latin squares, to the design of tournaments balanced for residual effects and to cryptography.
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