Let $A,B$ be monoids and $A\amalg B$ their product in the category of monoids, comprised of reduced words. Previously I have asked about the canonical arrow $A\amalg B\to A\times B$ given by e.g $$a_1a_2b_1a_3b_2b_3b_2a_2\mapsto(a_1a_2a_3a_2,b_1b_2b_3b_2).$$ This question is further along those lines. In catgories like unital mamgas and monoids the mentioned canonical arrow is always surjective. I am wondering about its injectivity.
The fiber of $(a,b)$ contains both $ab,ba$. Thus for this arrow to be injective these must coincide i.e $a,b$ commute in $A\amalg B$. Shouldn't there be no relations between elements of $A$ and $B$ in the coproduct? Does the equality $ab=ba$ imply $a$ or $b$ are the unit of $A$ or $B$? In particular, does the injectivity of the canonical arrow imply $A$ or $B$ is zero (trivial)?
The fiber of $(a_1a_2,b)$ contains for instance both $a_1ba_2$ and also $a_2ba_1$. Suppose these are equal. Since there are no relations between $A,B$ in the coproduct, can we cancel the $b$ and conclude $a_1a_2=a_2a_1$?
- Does the equality $a_1b=a_2b$ imply $a_1=a_2$?