_Monoids._

A monoid \$$(M,∗,e) \$$ is

1. a set \$$M \$$;

2. a function \$$∗:M×M \to M \$$; and

3. an element \$$e \in M \$$ called the identity;

subject to two laws:

Unit: the equations \$$e∗m=m \$$ and \$$m∗e=m \$$ hold for any \$$m \in M \$$.

Associative: the equation \$$(m1∗m2)∗m3=m1∗(m2∗m3) \$$ holds for any \$$m1,m2,m3 \in M \$$.

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(a) Show that \$$(\mathbb{N},+,0) \$$ forms a monoid.

(b) A string in 0 and 1 is a (possibly) empty sequence of 0s and 1s; examples include 0, 11, 0110, 0101110 and so on. We write the empty string []. Let \$$List_{0,1} \$$ be the set of strings in 0 and 1. Given two strings \$$a \$$ and \$$b \$$, we may concatenate them to form a new string \$$ab \$$. Show that \$$List_{0,1} \$$, together with concatenation and the empty string [], form a monoid.

(c) Explain why (prove that) every monoid can be viewed as a category with a single object.