  In the 1970s Deuber introduced the notion of $(m,p,c)$-sets in $\mathbb{N}$
and showed that these sets are partition regular and contain all linear
partition regular configurations in $\mathbb{N}$. In this paper we obtain
enhancements and extensions of classical results on $(m,p,c)$-sets in two
directions. First, we show, with the help of ultrafilter techniques, that
Deuber's results extend to polynomial configurations in abelian groups. In
particular, we obtain new partition regular polynomial configurations in
$\mathbb{Z}^d$. Second, we give two proofs of a generalization of Deuber's
results to general commutative semigroups. We also obtain a polynomial version
of the central sets theorem of Furstenberg, extend the theory of
$(m,p,c)$-systems of Deuber, Hindman and Lefmann and generalize a classical
theorem of Rado regarding partition regularity of linear systems of equations
over $\mathbb{N}$ to commutative semigroups.
