Word reduction and multiplication rules for forming product and coproduct of monoids and free monoids

Question

The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes.

$\color{Green}{Background:}$

$\textbf{(1)}$Rules for word reduction for Monoids

$\textbf{(1a)}$ Multiply adjacent pairs with the same $i-$term:

$(a_1,\ldots,a_j=(x_j,i),a_{j+1}=(x_{j+1},i),\ldots,a_n)\mapsto (a_1,\ldots,\hat{a}_j=(x_jx_{j+1},i),a_{j+2},\ldots,a_n)$

$\textbf{(1b)}$ Delete identity terms:

$(a_1,\ldots,a_j=(e_j,i_j),\ldots a_n)\mapsto (a_1,\ldots,a_{j-1},a_{j+1},\ldots,a_n)$

$\textbf{(2)}$ $\textbf{(Definition 2:)}$ The $\textbf{Free monoid on the set X of generators}$ is the set $X^*$ of all finite sequences of elements from $X$ (including the "empty" sequence $\Lambda$ of length $0$) with associative multiplication of $\textbf{concatenation}$

$$\textbf{(3)}\quad (x_1\ldots x_m)\cdot ({x_1}^{'}\ldots {x_n}^{'})=(x_1\ldots x_m, {x_1}^{'}\ldots {x_n}^{'})$$

for which $\Lambda$ is clearly the identity: $\Lambda\cdot w=w=w\cdot \Lambda$ for all $w\in X^*.$ Note that an element $x$ yields a string $(x)$ in $X^*$ of length one.

$\color{Red}{Questions:}$

Let $X=X_1\times X_2$ and $X'=X_1\sqcup X_2$ and suppose we are given strings, $(x_1\ldots x_m)$ and $({x_1}^{'}\ldots {x_n}^{'}),$ where $(x_1\ldots x_m)\in X_1$ and $({x_1}^{'}\ldots {x_n}^{'})\in X_2.$ I want to form either product or coproduct for both monoids and free monoids in the sense of category theory. I just have few quick questions. |

$1)$ The word reduction rules $\textbf{(1a)},$ $\textbf{(1b)},$ and multiplication rule $\textbf{(3)}$ can be apply to strings when forming either product or coproduct for both monoids and free monoids.

$2)$ For both monoids and free monoids, when forming their product and coproducts with strings $(x_1\ldots x_m)$ and $(x_1\ldots x_m),$ in the process, can we ever have the cases: $(x_1\ldots x_m)\cdot ({x_1}^{'}\ldots {x_n}^{'})\in X=X_1\times X_2$ or $(x_1\ldots x_m)\cdot ({x_1}^{'}\ldots {x_n}^{'})\in X'=X_1\sqcup X_2?$

Answer 1

$1)$ The word reduction rules $\textbf{(1a)},$ $\textbf{(1b)},$ and multiplication rule $\textbf{(3)}$ can be apply to strings when forming either product or coproduct for both monoids and free monoids.

The rules specifically apply to monoid coproducts. The product of monoids doesn't use words at all, just tuples with a componentwise multiplication.

$2)$ For both monoids and free monoids, when forming their product and coproducts with strings $(x_1\ldots x_m)$ and $(x_1\ldots x_m),$ in the process, can we ever have the cases: $(x_1\ldots x_m)\cdot ({x_1}^{'}\ldots {x_n}^{'})\in X=X_1\times X_2$ or $(x_1\ldots x_m)\cdot ({x_1}^{'}\ldots {x_n}^{'})\in X'=X_1\sqcup X_2?$

That depends entirely on what "$\cdot$" means. A priori, there is no definition of multiplication of elements of $X_1$ with elements of $X_2$ and neither the definition of product or coproduct defines such a thing. There are natural inclusions $X_i \to X_1 \times X_2$ and $X_i \to X_1 \sqcup X_2$. You could then define $\cdot$ by including each element into a common domain first.