PlanetMath: loop and quasigroup

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loop and quasigroup

(Definition)

A quasigroup is a groupoid with the property that for every $x, y \in G$ , there are unique elements $w, z \in G$ such that and .

A loop is a quasigroup which has an identity element.

What distinguishes a loop from a group is that the former need not satisfy the associative law.

"loop and quasigroup" is owned by mclase.

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Also defines:

loop, quasigroup

Attachments:: Moufang loop (Definition) by yark
medial quasigroup (Definition) by rspuzio

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Cross-references: associative, group, identity element, property, groupoid
There are 6 references to this entry.

This is version 1 of loop and quasigroup, born on 2002-09-06.
Object id is 3436, canonical name is LoopAndQuasigroup.
Accessed 7438 times total.

Classification:

20N05 (Group theory and generalizations :: Other generalizations of groups :: Loops, quasigroups)

Pending Errata and Addenda

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