Title: Extensions of Operators, Liftings of Monads and Distributive Laws,
Abstract: In a previous study, the algebraic formulation of the First Fundamental
Theorem of Calculus (FFTC) is shown to allow extensions of differential and
Rota-Baxter operators on the one hand, and to give rise to liftings of monads
and comonads, and mixed distributive laws on the other. Generalizing the FFTC,
we consider in this paper a class of constraints between a differential
operator and a Rota-Baxter operator. For a given constraint, we show that the
existences of extensions of differential and Rota-Baxter operators, of liftings
of monads and comonads, and of mixed distributive laws are equivalent. We
further give a classification of the constraints satisfying these equivalent
conditions.