Associator (mathematics)
In abstract algebra , the term associator is used in different ways as a measure of the deviation of an algebraic structure or a two-digit link defined there from the associative law.
Ring theory
In a non-associative ring or algebra , the associator is the multilinear map
where the spellings or are also common. For an (associative) ring it is always zero.
Identity applies to the associate
It alternates exactly when is alternative .
It is symmetric in its rightmost arguments when is a pre-Lie algebra .
Quasi-group theory
The associator in quasi-groups also measures the deviation of the link defined there from the associative law, but otherwise its definition differs fundamentally from that in ring theory.
A quasi-group Q is a set with the two-digit link for which the equations and have the unique solutions . In a quasi-group Q the associator is through
Are defined.
See also
literature
- Richard D. Schafer [1966]: An Introduction to Nonassociative Algebras . Dover, 1995, ISBN 0-486-68813-5 .