Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $\ast$ be a binary operation on a set $X$, let $m,n\in\mathbb{N}$, and let iterated operations be as in Iterated Operations: Finite Sums and Finite Products §iterated and Iterated Operations: Finite Sums and Finite Products §restriction.

¶ For $a:[n]\to X$, $\mathop{\ast}\limits_{k=1}^{1}a_{k}=a_{1}$; and for $a:[n+1]\to X$, $\mathop{\ast}\limits_{k=1}^{n+1}a_{k}=\Big(\mathop{\ast}\limits_{k=1}^{n}a_{k}\Big)\ast a_{n+1}$.

¶ Let $\diamond$ be a binary operation on a set $Y$ and $\varphi:X\to Y$ a map with $\varphi(x\ast y)=\varphi(x)\diamond\varphi(y)$ for all $x,y\in X$. For $a:[n]\to X$, $\varphi\Big(\mathop{\ast}\limits_{k=1}^{n}a_{k}\Big)=\mathop{\diamond}\limits_{k=1}^{n}\varphi(a_{k})$.

¶ If $\ast$ is associative, then for $a:[n+m]\to X$,

$$\mathop{\ast}\limits_{k=1}^{n+m}a_{k}=\Big(\mathop{\ast}\limits_{k=1}^{n}a_{k}\Big)\ast\Big(\mathop{\ast}\limits_{j=1}^{m}a_{n+j}\Big).$$

¶ If $\ast$ is associative and commutative, then for $a:[n]\to X$ and every bijection $\sigma:[n]\to[n]$, $\mathop{\ast}\limits_{k=1}^{n}a_{\sigma(k)}=\mathop{\ast}\limits_{k=1}^{n}a_{k}$.

¶ If $\ast$ is associative and commutative, then for $a,b:[n]\to X$, $\mathop{\ast}\limits_{k=1}^{n}(a_{k}\ast b_{k})=\Big(\mathop{\ast}\limits_{k=1}^{n}a_{k}\Big)\ast\Big(\mathop{\ast}\limits_{k=1}^{n}b_{k}\Big)$.