Number Systems Containing Non-Unique Additive Inverses

Question

I’ve seen proofs of the uniqueness of the additive inverse of a given element for specified number systems (e.g., the reals, fields in general, etc). Are there known instances of number systems in which a given element may have more than one additive inverse? Does the substitution property of equality preclude any system from containing more than one additive inverse for a given element?

Answer 1

Uniqueness of the inverse is a consequence of associativity. Suppose $*$ is an associative operation on the set $S$ and that $e$ is the neutral element for $*$, that is $$ e*x=x=x*e $$ for every $x\in S$. Define an inverse of $x\in S$ as an element $x'\in S$ such that $$ x*x'=e=x'*x $$ Then we can prove that if $x'$ and $x''$ are inverse element of $x$, then $x'=x''$.

Indeed $$ x'=x'*e=x'*(x*x'')=(x'*x)*x''=e*x''=x'' $$

With non associative operations, uniqueness of the inverse is not granted, it may or may not hold.

Answer 2

This property not only holds true for the reals and select fields, but for all groups. In any given group $G$, the inverse of a given element $a$ is unique.

If you were to take away the restrictions of a group, and define a binary algebraic structure where for arbitrary elements $a,b$ in a set $S$, we get $a+b = e$ for all $a,b \in S$, where $e$ is the identity element, one could argue that every element has multiple inverses.