Regular expression over a monoid
The sets that can be defined this way are called rational. 
As an example, the pairs of input-output strings $(u,v) \in \Sigma^*\times\Delta^*$ of accepted computations in finite state automata on two tapes define rational subsets in $\Sigma^*\times\Delta^*$ (with pair-wise concatenation as monoid operation). The word problem in $\Sigma^*\times\Delta^*$ is trivially decidable, but equality for their rational languages (of pairs of words) is not.

A special collection of monoids that were extensively studied for their rational language decision problems are the so-called trace monoids (or free partially commutative monoids). A special case of these monoids is $\Sigma^*\times\Delta^*$.

An overview of the literature in that area is given by Sakarovich, in his paper "The "last" decision problem for rational trace languages".