  We introduce a new algebraic construction, {\em monop}, that combines monoids
(with respect to the product of species), and operads (monoids with respect to
the substitution of species) in the same algebraic structure. By the use of
properties of cancellative set-monops we construct a family of partially
ordered sets whose prototypical examples are the Dowling lattices. They
generalize the partition posets associated to a cancellative operad, and the
subset posets associated to a cancellative monoid. Their generalized Withney
numbers of the first and second kind are the entries of a Riordan matrix and
its inverse. Equivalently, they are the connecting coefficients of two umbral
inverse Sheffer sequences with the family of powers $\{x^n\}_{n=0}^{\infty}$.
We study algebraic monops, their associated algebras and the free
monop-algebras, as part of a program in progress to develop a theory of Koszul
duality for monops.
