Loop

Definition
An algebra loop, sometimes simply called a loop, is defined as a set $$L$$ equipped with a binary operation $$*$$, such that the following are true:


 * neutral element: There exists an element $$e \in L$$ such that $$a * e = e * a = a \forall a \in l$$. Such an $$e$$ is termed a neutral element (or identity element or unit).
 * For any $$a,b \in L$$ there exist unique solutions $$x,y \in L$$ to the equation:

$$a * x = y * a = b$$

Equivalently, an algebra loop is a quasigroup with a neutral element (also called identity element or unit).

Stronger structures
For more, refer Category:Loop properties.

Weaker structures

 * Stronger than::Quasigroup
 * Stronger than::Cancellative magma

Left multiplication group
For an algebra loop $$L$$, every element $$x \in L$$ defines a permutation $$\lambda_x$$ on $$L$$ by $$\lambda_x(y) = x * y$$. The group generated by all these permutations is termed the left multiplication group. Note that the composite of $$\lambda_x$$ and $$\lambda_y$$ is not necessarily $$\lambda_{x * y}$$ because we are not assuming associativity.

Note that when $$L$$ is a group, the mapping $$x \mapsto \lambda_x$$ defines an isomorphism of groups, hence $$L$$ is isomorphic to its left multiplication group. Further, the action of $$L$$ (as the left multiplication group) on itself is the left regular representation.

Right multiplication group
For an algebra loop $$L$$, every element $$x \in L$$ defines a permutation $$\rho_x$$ on $$L$$ by $$\rho_x(y) = y * x$$. The group generated by all these permutations is termed the right multiplication group.

Note that when $$L$$ is a group, the mapping $$x \mapsto \rho_x$$ defines an anti-isomorphism of groups, hence $$L$$ is isomorphic to its right multiplication group. Further, the action of $$L$$ (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.