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Let $R$ be a factorial domain. In this work we investigate the connectionsbetween the arithmetic of ${\rm Int}(R)$ (i.e., the ring of integer-valuedpolynomials over $R$) and its monadic submonoids (i.e., monoids of the form$\{g\in {\rm Int}(R)\mid g\mid_{{\rm Int}(R)} f^k$ for some$k\in\mathbb{N}_0\}$ for some nonzero $f\in {\rm Int}(R)$). Since every monadicsubmonoid of ${\rm Int}(R)$ is a Krull monoid it is possible to describe thearithmetic of these monoids in terms of their divisor-class group. We give anexplicit description of these divisor-class groups in several situations andprovide a few techniques that can be used to determine them. As an applicationwe show that there are strong connections between ${\rm Int}(R)$ and itsmonadic submonoids. If $R=\mathbb{Z}$ or more generally if $R$ has sufficientlymany "nice" atoms, then we prove that the infinitude of the elasticity and thetame degree of ${\rm Int}(R)$ can be explained by using the structure ofmonadic submonoids of ${\rm Int}(R)$.

On the divisor-class group of monadic submonoids of rings of integer-valued polynomials