### 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) |
---|---|

Pages (from-to) | 209-215 |

Number of pages | 7 |

Journal | Commentationes Mathematicae Universitatis Carolinae |

Volume | 51 |

Issue number | 2 |

State | Published - Jan 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Commentationes Mathematicae Universitatis Carolinae*,

*51*(2), 209-215.

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*Commentationes Mathematicae Universitatis Carolinae*, vol. 51, no. 2, pp. 209-215.

**Right division in Moufang loops.** / Giuliani, Maria de Lourdes M.; Johnson, Kenneth.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Right division in Moufang loops

AU - Giuliani, Maria de Lourdes M.

AU - Johnson, Kenneth

PY - 2010/1/1

Y1 - 2010/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85068263208&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068263208&partnerID=8YFLogxK

M3 - Article

VL - 51

SP - 209

EP - 215

JO - Commentationes Mathematicae Universitatis Carolinae

JF - Commentationes Mathematicae Universitatis Carolinae

SN - 0010-2628

IS - 2

ER -