Loop
This is a variation of group|Find other variations of group | Read a survey article on varying group
This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.
View other such properties
QUICK PHRASES: group without associativity, unital magma with unique left and right quotients
Contents
Definition
An algebra loop, sometimes simply called a loop, is defined as a set equipped with a binary operation
, such that the following are true:
- neutral element: There exists an element
such that
. Such an
is termed a neutral element (or identity element or unit).
- For any
there exist unique solutions
to the equation:
Equivalently, an algebra loop is a quasigroup with a neutral element (also called identity element or unit).
Relation with other structures
Stronger structures
Structure | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
group | associative | loop not implies group | Alternative loop, Power-associative loop|FULL LIST, MORE INFO | |
Moufang loop | Alternative loop, Power-associative loop|FULL LIST, MORE INFO | |||
Alternative loop | |FULL LIST, MORE INFO |
For more, refer Category:Loop properties.
Weaker structures
Groups associated to an algebra loop
Left multiplication group
For an algebra loop , every element
defines a permutation
on
by
. The group generated by all these permutations is termed the left multiplication group. Note that the composite of
and
is not necessarily
because we are not assuming associativity.
Note that when is a group, the mapping
defines an isomorphism of groups, hence
is isomorphic to its left multiplication group. Further, the action of
(as the left multiplication group) on itself is the left regular representation.
Right multiplication group
For an algebra loop , every element
defines a permutation
on
by
. The group generated by all these permutations is termed the right multiplication group.
Note that when is a group, the mapping
defines an anti-isomorphism of groups, hence
is isomorphic to its right multiplication group. Further, the action of
(as the right multiplication group) on itself is the right regular representation.
Left inner mapping group
The left inner mapping group for an algebra loop is the subgroup of the left multiplication group, comprising those elements that send the identity element to itself.
When the algebra loop is a group, the left inner mapping group is the trivial subgroup.
Right inner mapping group
The right inner mapping group for an algebra loop is the subgroup of the right multiplication group, comprising those elements that send the identity element (of the loop) to itself.