It is accepted that two elements are inverse to each other if their product is equal to the identity element:
Inverse element in a magma
https://en.wikipedia.org/wiki/Inverse_element
The definition works fine for associative structures.
But let us take a closer look at non-associative case.
What do we mean by inverse element?
Intuitively, it is an element that reverses the action of a given element.
And what is the action of an element?
The most appropriate definition of an action of an element $a$ onto a magma $A(\cdot)$ is its translation $a \cdot A$ or $A \cdot a$.
But then the inverse action should be defined as $(a \cdot A)^{-1}$ or $(A \cdot a)^{-1}$.
In other words, a left inverse element of an element $x$ is an element $y$ such that $y \cdot A = (x \cdot A)^{-1}$.
A right inverse element of an element $x$ is an element $y$ such that $A \cdot y = (A \cdot x)^{-1}$.
On element level the definitions can be written as:
- $y$ is a left inverse element of $x$ if $y \cdot (x \cdot a) = x \cdot (y \cdot a) = a$ for any element $a$ of $A(\cdot)$.
- $y$ is a right inverse element of $x$ if $(a \cdot x) \cdot y = (a \cdot y) \cdot x = a$ for any element $a$ of $A(\cdot)$.
Obviously, in order for the inverse translations $(a \cdot A)^{-1}$ or $(A \cdot a)^{-1}$ to exist,
the actions $a \cdot A$ or $A \cdot a$ must be permutations.
These definitions look more consistent and do not require existence of the identity element,
similar to how the definition of an inverse function does not require existence of an identity function.
For example, all elements are inverse to themselves in the following quasigroup:
$$ \begin{array}{c|ccc} \cdot & a & b & c \\ \hline a & a & c & b \\ b & c & b & a \\ c & b & a & c \end{array}$$
Indeed, each element of the magma reverses its own action:
$a \cdot (a \cdot a) = a$, $a \cdot (a \cdot b) = b$, $a \cdot (a \cdot c) = c$, etc.
For associative structures the definitions coincide:
$y \cdot (x \cdot a) = (y \cdot x) \cdot a = a$ for any element $a$ if and only if $y \cdot x = e$.
The new definitions establish the same connection between the inverse operations (divisions) and the inverse elements in a quasigroup as in a group:
An element $x^{-1}$ of a quasigroup is a left (right) inverse of an element $x$ if an only if for any element $a$ of the quasigroup
- $x \backslash a = x^{-1} \cdot a$ (for the left inverse)
- $a / x = a \cdot x^{-1}$ (for the right inverse)
where $\backslash$ and $/$ are the left and the right inverse operations respectively.
Does it make sense to switch to the new definitions for all algebraic structures?