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Maps such as $\eta$ and $\xi$ are called measure-preserving and are studied in ergodic theory. In particular ergodic theory views these as dynamical systems, because the maps can be iterated. One th …

An algebraic structure such as a group, ring, field, etc. is usually defined to be a set with some operations satisfying certain properties. I am curious what, if anything, goes wrong when the underl …

It is at least sometimes called a "monoid semiring" by analogy with "group ring". As such it would be notated $S = \mathbb{N_0}[M]$ (or $\mathbb{N}[M]$ depending how you define things). By the way, …

Building on Ralph's answer a bit we can get uncountably many inequivalent examples as Mark Grant's comment on the original post suggested there should be. Let $S,T$ be a partition of the primes into …

$z\mapsto \frac{1}{z}$

As written the statement is false for $n=3$: note that $p_3(2,2) = 0$ but $p_3(3,0) > 0$, while $|(2,2)| < |(3,0)|$. Similar counterexamples exist for all $n\geq 5$. So for larger $n$ you would at l …

Using the axiom of choice, one can show that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as additive groups. In particular, they are both vector spaces over $\mathbb{Q}$ and AC gives bases of thes …