Monoid kernel

Set

 context $X$ context $M$ … monoid context $f:X\to M$
 definiendum $x\in\mathrm{ker}(f)$
 postulate $f(x)=e$

Discussion

$\mathrm{ker}(f)$ is $\mathrm{sol}(f)$ w.r.t. the unit of the monoid.

Reference

Wikipedia: Solution set