Coproduct of $(0,1)$-Algebras

Question

I am trying to find the coproduct of $(\mathbb {Z},0,+1) $ with itself in the category of $(0,1) $-Algebras. Finding $\mathbb {N}\sqcup\mathbb {N} $ was easy, since $\mathbb{N} $ is initial. But I don't know how this coproduct looks in general.

Answer 1

Coproducts can be computed by means of generators and relations. In this case, it is not hard to see that the coproduct in question is two copies of $\mathbb{Z}$ glued along the non-negative integers, i.e. the algebra whose underlying set is $(\mathbb{Z} \times \{ 0, 1 \}) / \sim$ where $(n, m) \sim (n', m')$ if and only if $n = n'$ and either $m = m'$ or $n \ge 0$, with the distinguished constant being $(0, 0)$ and the operation being $(n, m) \mapsto (n + 1, m)$.