Right division in Moufang loops

Maria de Lourdes M. Giuliani, Kenneth Johnson

Research output: Contribution to journalArticle

Abstract

If (G, ·) is a group, and the operation (*) is defined by x * y = x · y-1 then by direct verification (G, *) is a quasigroup which satisfies the identity (x * y) * (z * y) = x * z. Conversely, if one starts with a quasigroup satisfying the latter identity the group (G, ·) can be constructed, so that in effect (G, ·) 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 replacing loop multiplication by right division give identities in which loop inverses appear. However, it is possible with further work to obtain an identity in terms of (*) alone. The construction of the Moufang loop from a quasigroup satisfying this identity is significantly more difficult than in the group case, and it was first carried out using the software Prover9. Subsequently a purely algebraic proof of the construction was obtained.

Original language English (US) 209-215 7 Commentationes Mathematicae Universitatis Carolinae 51 2 Published - Jan 1 2010

Moufang Loop
Division
Quasigroup
Multiplication
Software

All Science Journal Classification (ASJC) codes

• Mathematics(all)

Cite this

Giuliani, Maria de Lourdes M. ; Johnson, Kenneth. / Right division in Moufang loops. In: Commentationes Mathematicae Universitatis Carolinae. 2010 ; Vol. 51, No. 2. pp. 209-215.
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Right division in Moufang loops. / Giuliani, Maria de Lourdes M.; Johnson, Kenneth.

In: Commentationes Mathematicae Universitatis Carolinae, Vol. 51, No. 2, 01.01.2010, p. 209-215.

Research output: Contribution to journalArticle

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