Let $\sim$ be an equivalence relation over an algebraic object $X$ whereby there is a modular algebra $X/{\sim},\star$.
Does there already exist a concept ever so slightly weaker than torsion, whereby if $[x]$ is a representative of some element of $X/{\sim}$ then $\lvert [x]\star [x]\rvert=\lvert [x]\rvert$, but for all distinct $[x_1],[x_2]\in X :\lvert[x_1]\star[x_2]\rvert<\max\{\lvert[x_1]\rvert,\lvert[x_2]\rvert\}$
I've not defined $\lvert\cdot\rvert$ which would take a little work but this measures something analogous to the order in a group.
An example of this property applies to the multiplicative monoid of odd, positive numbers (with each odd number $[x]$ a representative of $\langle2\rangle[x]$). In this case (nonassociative) addition of the representatives is the modular operation with the required property. In this example we have e.g. $26\star 44=[13+11]=3$