Free Monoids

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Exercises

1. You might think (as I did, originally) that the requirement that a homomorphism of monoids preserve the unit is redundant. After all, we know that for all $a$

h a * h e = h (a * e) = h a

So $he$ acts like a right unit (and, by analogy, as a left unit). The problem is that $ha$, for all $a$ might only cover a sub-monoid of the target monoid. There may be a “true” unit outside of the image of $h$. Show that an isomorphism between monoids that preserves multiplication must automatically preserve unit.

2. Consider a monoid homomorphism from lists of integers with concatenation to integers with multiplication. What is the image of the empty list []? Assume that all singleton lists are mapped to the integers they contain, that is [3] is mapped to 3, etc. What’s the image of [1, 2, 3, 4]? How many different lists map to the integer 12? Is there any other homomorphism between the two monoids?
3. What is the free monoid generated by a one-element set? Can you see what it’s isomorphic to?