union of a chain of monoids as monoid

1. Proposition

Let $M₁ \subseteq M₂ … $ be a chain of monoids. Then $M \coloneqq \bigcup_{n \in \mathbb{N}} M_n$ is also a monoid

2. Proof

Let $m_1,m_2 \in M$, then by assumption there exists $M_m,M_n$ such that $m_1 \in M_m, m_2 \in M_n$. Hence $m_1,m_2 \in M_{\mathrm{max}(m,n)}$ and thus

2.1. closed

\begin{align*} m_1 \cdot m_2 \in M_{\mathrm{max}(m,n)} \subseteq M \end{align*}

2.2. associative

For $m_3$, w.l.o.g. $m_3 \in M_{\mathrm{max}(m,n)}$ we conclude

\begin{align*} m_1 \cdot (m_2 \cdot m_3) = (m_1 \cdot m_2) \cdot m_3 \end{align*}