composition of monoid homomorphisms as monoid homomorphism

1. Proposition

Let \(M_1,M_2,M_3\) be monoids and \(\varphi_1: M_1 \rightarrow M_2\) and \(\varphi_2: M_2 \rightarrow M_3\) monoid-homomorphism. Then the composition

\begin{align*} \varphi_2 \circ \varphi_1: M_1 \rightarrow M_3 \end{align*}

is also a monoid homomorphism

2. Proof

2.1. unit

for \(1_{M_1}\), it follows by assumption, that

\begin{align*} \varphi_2 \circ \varphi_1 (1_{M_1}) =& \varphi_2 ( 1_{M_2}) \\ =& 1_{M_3} \end{align*}

2.2. additivity

Let \(m_1,m_2 \in M\), then

\begin{align*} \varphi_2 \circ \varphi_1 (m_1 \cdot m_2) =& \varphi_2 (\varphi_1(m_1) \cdot \varphi_1(m_2)) \\ =& \varphi_2(\varphi_1(m_1)) \cdot \varphi_2(\varphi_1(m_2)) \end{align*}