A right-invertible and left-invertible operation

abstract-algebra

im reading a book and it says:

  • An operation O is right invertible or left invertible in the set K if for any two elements x and y of the set K there always exists an element z of K such that x=yOz or x=zOy.

  • An operation O which is both right and left invertible is simply invertible in the class K.

  • K is a group with respect to O if this K is closed under O and O is associative and invertible in K.

Wikipedia says:

"To qualify as a group the set and operation, (G, •), must satisfy four requirements:

  • Closure:
    For all a, b in G, the result of the operation, a•b, is also in G.

  • Associativity:
    For all a, b and c in G, (a•b)•c = a•(b • c).

  • Identity element:
    There exists an element e in G such that, for every element a in G, the equation e•a=a•e=a holds.

  • Inverse element:
    For each a in G, there exists an element b in G, such that a•b=b•a=e, where e is the identity element."

I don't really get it, the closure property and the associativity property are the same in both definitions of group, but how right invertible and left invertible are the same of "identity element" or "inverse element"?

Best Answer

In your post there are two concepts that are strictly linked but not the same...

One is that invertible operation, while the other is that of inverse element of an algebraic structure.

For e.g. a group $G$, the second one refer to the operation "$\circ"$ defined on the structure, but in addition needs the existence of a "neutral (or: identical) element" $e$:

"An element of the algebraic structure $G$ with a two-sided inverse in $G$ is called invertible in $G$."

See some useful schema regarding the classification of algebraic structures:

Magma, Semigroup, Monoid, etc.

Related Question