What's the name for the property for which $x + x = 0 \Longleftrightarrow x = 0$? I have a set $\mathbb{S}$ for which I have defined an operation: addition ($+ : \mathbb{S} \times \mathbb{S} \rightarrow \mathbb{S}$). The structure $(\mathbb{S}, +)$ is a group.
I have shown that if $x + x = 0$, then $x$ is not necessarily $0$ (where $x \in \mathbb{S}$ and $0$ is the identity element).
Clearly, my $+$ operator is lacking a property. How is this property called?
 A: You might want to say that $(\mathbb{S}, +)$ does not have $2$-torsion, or perhaps that it is $2$-torsion-free.
A: An element $x$ such that $x+x=0$ is called an involution. See here under the "group theory" heading:
https://en.m.wikipedia.org/wiki/Involution_(mathematics)
This is the opposite of the property you describe, and seems to be the rarer case. An operation under which no element has order 2 could perhaps be called "involution-free"
A: If your set $\mathbb{S}$ is a ring, such property perhaps could be stated as : $characteristic(\mathbb{S})\not=2$
wikipedia: Characteristic (algebra)
A: In a group $G$ ,written multiplicatively,with identity element $1$: For $x\in G,$ if $x^n=1$ for some positive integer $n,$ then the least such $n$ is called the order of $x$ (or the order of $x$ in $G$), sometimes written $[x]=n.$ The order need not exist. For example in the additive group of Reals, only the identity $0$ has an order : $[0]=1.$  You can say, for your Q, that no member of $S$ has order $2.$
