  The algebraic expression $3 + 2 + 6$ can be evaluated to $11$, but it can
also be partially evaluated to $5 + 6$. In categorical algebra, such partial
evaluations can be defined in terms of the $1$-skeleton of the bar construction
for algebras of a monad. We show that this partial evaluation relation can be
seen as the relation internal to the category of algebras generated by relating
a formal expression to its total evaluation. The relation is transitive for
many monads which describe commonly encountered algebraic structures, and more
generally for BC monads on $\mathsf{Set}$ (which are those monads for which the
underlying functor and the multiplication are weakly cartesian). We find that
this is not true for all monads: we describe a finitary monad on $\mathsf{Set}$
for which the partial evaluation relation on the terminal algebra is not
transitive. With the perspective of higher algebraic rewriting in mind, we then
investigate the compositional structure of the bar construction in all
dimensions. We show that for algebras of BC monads, the bar construction has
fillers for all directed acyclic configurations in $\Delta^n$, but generally
not all inner horns.
