### Locally flat embeddings of Hilbert cubes are flat

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Let M be a commutative cancellative monoid. The set Δ(M), which consists of all positive integers which are distances between consecutive factorization lengths of elements in M, is a widely studied object in the theory of nonunique factorizations. If M is a Krull monoid with cyclic class group of order n ≥ 3, then it is well-known that Δ(M) ⊆ {1,..., n-2}. Moreover, equality holds for this containment when each class contains a prime divisor from M. In this note, we consider the question of determining...

Let ℕ represent the positive integers and ℕ₀ the non-negative integers. If b ∈ ℕ and Γ is a multiplicatively closed subset of ${\mathbb{Z}}_{b}=\mathbb{Z}/b\mathbb{Z}$, then the set ${H}_{\Gamma}=x\in \mathbb{N}|x+b\mathbb{Z}\in \Gamma \cup 1$ is a multiplicative submonoid of ℕ known as a congruence monoid. An arithmetical congruence monoid (or ACM) is a congruence monoid where Γ = ā consists of a single element. If ${H}_{\Gamma}$ is an ACM, then we represent it with the notation M(a,b) = (a + bℕ₀) ∪ 1, where a, b ∈ ℕ and a² ≡ a (mod b). A classical 1954 result of James and Niven implies that the only ACM...

If $a$ and $b$ are positive integers with $a\le b$ and ${a}^{2}\equiv a\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}b$, then the set
$${M}_{a,b}=\{x\in \mathbb{N}:x\equiv a\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}b\phantom{\rule{4pt}{0ex}}\text{or}\phantom{\rule{4pt}{0ex}}x=1\}$$
is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid $M$ with units ${M}^{\times}$ and any $x\in M\setminus {M}^{\times}$ we say that $t\in \mathbb{N}$ is a

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