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

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Let R be a factorial domain. In this work we investigate the connections between the arithmetic of Int(R) (i.e., the ring of integer-valued polynomials over R) and its monadic submonoids (i.e., monoids of the form {g∈ Int(R) g Int(R) fᵏ for some k₀} for some nonzero f∈ Int(R)). Since every monadic submonoid of Int(R) is a Krull monoid it is possible to describe the arithmetic of these monoids in terms of their divisor-class group. We give an explicit description of these divisor-class groups in several situations and provide a few techniques that can be used to determine them. As an application we show that there are strong connections between Int(R) and its monadic submonoids. If R=Z or more generally if R has sufficiently many "nice" atoms, then we prove that the infinitude of the elasticity and the tame degree of Int(R) can be explained by using the structure of monadic submonoids of Int(R).