Moufang loop: Difference between revisions
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==Definition== | ==Definition== | ||
===In terms of Moufang's identities=== | |||
A '''Moufang loop''' is an [[algebra loop]] <math>L</math> with multiplication <math>*</math> satisfying the following three identities: | A '''Moufang loop''' is an [[algebra loop]] <math>L</math> with multiplication <math>*</math> satisfying the following three identities: | ||
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# <math>\! (z * x) * (y * z) = (z * (x * y)) * z \ \forall \ x,y,z \in L</math> | # <math>\! (z * x) * (y * z) = (z * (x * y)) * z \ \forall \ x,y,z \in L</math> | ||
===In terms of Bol loops=== | |||
A '''Moufang loop''' is an [[algebra loop]] that is both a [[defining ingredient::left Bol loop]] and a [[defining ingredient::right Bol loop]]. | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 23:18, 5 March 2010
This article defines a property that can be evaluated for a loop.
View other properties of loops
Definition
In terms of Moufang's identities
A Moufang loop is an algebra loop with multiplication satisfying the following three identities:
In terms of Bol loops
A Moufang loop is an algebra loop that is both a left Bol loop and a right Bol loop.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Group | an associative loop (see nonempty associative quasigroup equals group) | |||
| Finite Moufang loop |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Alternative loop | algebra loop satisfying the left-alternative and right-alternative identities | Moufang implies alternative | alternative not implies Moufang | |FULL LIST, MORE INFO |