Suppose we are given the finite cyclic group $\mathbb{Z}/b\mathbb{Z}$ and the monoid of natural numbers $\mathbb{N}$, both of which are semigroups. Does the restricted wreath product $(\mathbb{Z}/b\mathbb{Z}) \wr \mathbb{N}$ define the base-$b$ positional notation system for natural numbers?
2026-03-25 06:12:00.1774419120
Are positional notation systems for natural numbers wreath products of semigroups?
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No, because the wreath product does not implement carrying as necessary for a positional notation system. It would instead define an group structure isomorphic to the polynomial ring over the ring of integers modulo $b$.