  Weakly Schreier split extensions are a reasonably large, yet well-understood
class of monoid extensions, which generalise some aspects of split extensions
of groups. This short note provides a way to define and study similar classes
of split extensions in general algebraic structures (parameterised by a term
$\theta$). These generalise weakly Schreier extensions of monoids, as well as
general extensions of semi-abelian varieties (using the $\theta$ appearing in
their syntactical characterisation). Restricting again to the case of monoids,
a different choice of $\theta$ leads to a new class of monoid extensions, more
general than the weakly Schreier split extensions.
