  The model of programs over (finite) monoids, introduced by Barrington and
Th\'erien, gives an interesting way to characterise the circuit complexity
class $\mathsf{NC^1}$ and its subclasses and showcases deep connections with
algebraic automata theory. In this article, we investigate the computational
power of programs over monoids in $\mathbf{J}$, a small variety of finite
aperiodic monoids. First, we give a fine hierarchy within the class of
languages recognised by programs over monoids from $\mathbf{J}$, based on the
length of programs but also some parametrisation of $\mathbf{J}$. Second, and
most importantly, we make progress in understanding what regular languages can
be recognised by programs over monoids in $\mathbf{J}$. To this end, we
introduce a new class of restricted dot-depth one languages, threshold
dot-depth one languages. We show that programs over monoids in $\mathbf{J}$
actually can recognise all languages from this class, using a non-trivial
trick, and conjecture that threshold dot-depth one languages with additional
positional modular counting suffice to characterise the regular languages
recognised by programs over monoids in $\mathbf{J}$. Finally, using a result by
J. C. Costa, we give an algebraic characterisation of threshold dot-depth one
languages that supports that conjecture and is of independent interest.
