Quasigroups

A quasigroup is a groupoid (S,*) that guarantees the existence of both a right inverse (dx) and a left inverse (sx) for every pair of elements a and b, satisfying the equations $$ a*d_x=b $$ and $$ s_x*a=b $$.

Quasigroups belong to a broader category of groupoids.

This means every quasigroup is a groupoid, but does not necessarily have commutative inverses.

Example of a quasigroup

If a quasigroup also contains the neutral element for its operation, it is known as a loop.

    An Applied Example

    The set of all integers Z under the operation of subtraction (-) forms a quasigroup (Z,-).

    $$ (Z,-) $$

    We will now review the properties of this algebraic structure.

    Subtraction is a closed operation within the set of integers Z, as the difference between any two integers results in another integer.

    $$ \forall a,b \in Z \Longrightarrow a-b \in Z $$

    Thus, the structure (Z,-) qualifies as a groupoid.

    Example: Consider integers a=4 and b=7; their difference is $$ a-b=4-7=-3 $$, another integer.

    Every pair of integers a and b has both a right and a left inverse.

    Consequently, the structure (Z,-) is a quasigroup.

    Example: For the integers a=4 and b=7, the right inverse x=-3 fulfills $$ a-x = b $$ $$ 4-x = 7 $$ $$ x = 4-7 $$ $$ x=-3 $$, while the left inverse y=11 satisfies $$ y-a=b $$ $$ y-4=7 $$ $$ y-4=7 $$ $$ y=7+4 $$ $$ y=11 $$.

    And so forth.

     
     

    Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

    FacebookTwitterLinkedinLinkedin
    knowledge base

    Abstract Algebra