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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.
Le système AutoGraphiX (AGX1 et AGX2) permet, parmi d’autres fonctions, la génération automatique de conjectures en théorie des graphes et, dans une version plus récente, la preuve automatique de conjectures simples. Afin d’illustrer ces fonctions et le type de résultats obtenus, nous étudions systématiquement ici des conjectures obtenues par ce système et de la forme où désigne la maille (ou longueur du plus petit cycle) du graphe , un autre invariant choisi parmi le nombre de stabilité,...
Le système AutoGraphiX (AGX1 et AGX2) permet,
parmi d'autres fonctions, la génération automatique de conjectures en
théorie des graphes et, dans une version plus récente, la preuve automatique de conjectures simples. Afin
d'illustrer ces fonctions et le type de résultats obtenus, nous étudions systématiquement ici des conjectures
obtenues par ce système et de la forme où g désigne la maille (ou longueur
du plus petit cycle) du graphe G=(V, E), i un autre invariant choisi
parmi le nombre...
In the field of automatic proving, the study of the sets of prime implicants or implicates of a formula has proven to be very important. If we focus on non-classical logics and, in particular, on temporal logics, such study is useful even if it is restricted to the set of unitary implicants/implicates [P. Cordero, M. Enciso, and I. de Guzmán: Structure theorems for closed sets of implicates/implicants in temporal logic. (Lecture Notes in Artificial Intelligence 1695.) Springer–Verlag, Berlin 1999]....
If is a group, and the operation is defined by then by direct verification is a quasigroup which satisfies the identity . Conversely, if one starts with a quasigroup satisfying the latter identity the group can be constructed, so that in effect is determined by its right division operation. Here the analogous situation is examined for a Moufang loop. Subtleties arise which are not present in the group case since there is a choice of defining identities and the identities produced by...
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