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MIT PRIMES/Art of Problem Solving

CROWDMATH 2022: Factorizations in Additive Structures

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Solution to Problem 7.1
Stiffler   2
N Feb 3, 2023 by Stiffler
Hello!! In a recent post Julmath presented an example that solve part (a) of Problem 7.1. Then I will focus in solve part (b).

Solution of part (b)
2 replies
Stiffler
Jan 3, 2023
Stiffler
Feb 3, 2023
Proposition Problem 4
Stiffler   1
N Jan 2, 2023 by felixgotti
In the solution of open problem $5$ Banghenz found and $\alpha > 1$ such that $\mathbb{N}_0[\alpha]$ is a non-FGM monoid with $\mathsf{c}(\mathbb{N}_0[\alpha]) = \infty$. In a recent post Julmath presents an example, again with $\alpha > 1$, where $\mathsf{c}(\mathbb{N}_0[\alpha]) < \infty$. Here I address a result that I found that contrasts with their examples since it focuses in the case $\alpha < 1$.

Proposition: Let $n > 2$ be a positive integer. For every $d > 1$ there exists a non-FGM rank-d monoid $\mathbb{N}_0[\alpha]$ with $\alpha < 1$ and $\mathsf{c}(x) = n$ for every $x \in M$.

Proof
1 reply
Stiffler
Dec 2, 2022
felixgotti
Jan 2, 2023
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