union of a chain of commutative monoids as commutative monoid
1. Proposition
Let $M1 \subseteq W2 …$ be a chain of commutative monoids Then \(M \coloneqq \bigcup_{n \in \mathbb{N}} M_n\) is also a commutative monoid.
2. Proof
2.2. commutative
Let \(m_1,m_2 \in G\), then by construction there exist \(M_n,M_m\) with \(m_1 \in M_m, m_2 \in M_n\). Hence \(m_1,m_2 \in M_{\mathrm{max}(m,n)}\) and thus
\begin{align*} m_1 \cdot m_2 = m_2 \cdot m_1 \end{align*}