Recall the following definitions:
- The Boolean monoid is the monoid $\mathbb{B}=(\{0,1\},\max,0)$;
- The free semiring on a commutative monoid $A$ is the semiring $\mathrm{Free}(A)$ consisting of
- The Underlying Commutative Monoid. The commutative monoid $\mathrm{Free}(A)_+$ defined by \begin{align*} \mathrm{Free}(A)_+ &\overset{\mathrm{def}}{=} \bigoplus_{n=0}^{\infty}A^{\otimes_{\mathbb{N}}n}\\ &=\mathbb{N}\oplus A\oplus(A\otimes_{\mathbb{N}}A)\oplus(A\otimes_{\mathbb{N}}A\otimes_{\mathbb{N}}A)\oplus\cdots; \end{align*}
- The Multiplication. The map
$$\mathrm{Free}(A)_+\otimes_{\mathbb{N}}\mathrm{Free}(A)_+\to\mathrm{Free}(A)_+$$
obtained as follows:
- First we write $$\mathrm{Free}(A)_+\otimes_{\mathbb{N}}\mathrm{Free}(A)_+\cong\bigoplus_{k\in\mathbb{N}}\bigoplus_{\substack{n,m\in\mathbb{N}\\n+m=k}}A^{\otimes_{\mathbb{N}}k};$$
- Then we iterate the unit and multiplication morphisms \begin{align*} \eta_{A} &\colon \mathbb{N} \to A,\\ \mu_{A} &\colon A\oplus A \to A \end{align*} of $A$ to obtain morphisms of the form $(A^{\otimes_{\mathbb{N}}k})^{\oplus k+1}\to A^{\otimes_{\mathbb{N}}k}$;
- Finally we use these morphisms to get our desired multiplication map $$\underbrace{\bigoplus_{k\in\mathbb{N}}\bigoplus_{\substack{n,m\in\mathbb{N}\\n+m=k}}A^{\otimes_{\mathbb{N}}k}}_{\overset{\mathrm{def}}{=}\mathrm{Free}(A)_+\otimes_{\mathbb{N}}\mathrm{Free}(A)_+}\to\underbrace{\bigoplus_{k\in\mathbb{N}}A^{\otimes_{\mathbb{N}}k}}_{\overset{\mathrm{def}}{=}\mathrm{Free}(A)_+}.$$
Question. Does the semiring $\mathrm{Free}(\mathbb{B})$ admit a nice, simple description? Since $\mathbb{B}\otimes_{\mathbb{N}}\mathbb{B}\cong\mathbb{B}$, it follows that the underlying commutative monoid of $\mathrm{Free}(\mathbb{B})$ is simply $\mathbb{N}\oplus\mathbb{B}\oplus\mathbb{B}\oplus\cdots$, and thus the elements of $\mathrm{Free}(\mathbb{B})$ look like strings of the form $(n,[\text{0 or 1}],[\text{0 or 1}],\cdots)$, with one example of addition in $\mathrm{Free}(\mathbb{B})$ being $$(n,1,0,1,0,0,\cdots)+(m,1,1,0,0,\cdots)=(n+m,1,1,1,0,0,\cdots).$$ Multiplication is slightly more cumbersome to compute, but is also defined in somewhat a simple way.
The reason I'm curious about this is that I was wondering whether we could somehow relate $\mathrm{Free}(\mathbb{B})$ to the tropical semiring $(\mathbb{N}\cup\{-\infty\},\max,-\infty)$, as we can also write $\mathbb{B}$ like $(\{-\infty,1\},\max,-\infty)$, and then "freely generating addition of $1$ with itself" (as we do when moving from $\mathrm{pt}=\{1\}$ to the natural numbers $\mathbb{N}$ by taking the free commutative monoid on $\mathrm{pt}$) should hopefully recover $\mathbb{N}\cup\{-\infty\}$.