# Examples of prime spectra of monoids

Given a (pointed or unpointed) monoid $$M$$, one defines an ideal of $$M$$ to be a subset $$I$$ of $$M$$ such that if $$a\in I$$ and $$r\in M$$, then $$ra\in I$$ (and such that $$0\in I$$ if $$M$$ is pointed). A prime ideal $$\frak p$$ of $$M$$ is then an ideal such that if $$ab\in\frak p$$, then $$a\in\frak p$$ or $$b\in\frak p$$. The set of all prime ideals of $$M$$ can is written $$\mathrm{MSpec}(M)$$, and it can be made into a topological space equipped with a sheaf of monoids, giving a variant of algebraic geometry where monoids play the role of rings. Good references for this are Sections 1--2 of this paper and Ogus's book.

Lately I've been trying to compile some examples of such prime spectra, and wondering about how these relate to the usual $$\mathrm{Spec}$$ of a ring. So far, I've found or read about the following examples (the first five examples below come from Martin Brandenburg's answer here):

• $$\mathrm{MSpec}(\mathbf{N},+)=\{0,\mathbf{N}_{>0}\}$$.
• $$\mathrm{MSpec}(\mathbf{Z},+)=\{0\}$$.
• $$\mathrm{MSpec}(\mathbf{N},\cdot)=\mathrm{MSpec}(\mathbf{Z},\cdot)=\mathcal{P}(\mathbf{P})$$, the powerset of the set $$\mathbf{P}$$ of all prime numbers.
• $$\mathrm{MSpec}(\mathbf{N}\otimes_{\mathbf{N}_+}\mathbf{N})\cong\mathcal{P}(\mathbf{P}\times\mathbf{P})$$.
• $$\mathrm{MSpec}(\mathbf{Z}\otimes_{\mathbf{F}_1}\mathbf{Z})\cong\mathcal{P}(\mathbf{P})\times\mathcal{P}(\mathbf{P})$$.
• $$\mathrm{MSpec}(K)=\{(0)\}$$ whenever $$K^\times=K\setminus\{0\}$$. In particular this applies to $$\mathbf{F}_1=\{0,1\}$$.
• $$\mathrm{MSpec}(\mathbf{F}_1[x])=\{(0),(x)\}$$.
• $$\mathrm{MSpec}(\mathbf{F}_1[x,y])=\{(0),(x),(y),(x,y)\}$$.
• $$\mathrm{MSpec}(\mathbf{F}_1[x_1,...,x_n])=\mathcal{P}(\{x_1,...x_n\})$$.
• $$\mathrm{MSpec}(\mathbf{F}_1[t,t^{-1}])=\{(0),(t),(t^{-1})\}$$.

Now, any ring $$R$$ has an associated monoid, given by keeping only multiplication, and these should give more examples. For the usual prime spectrum of a ring, we have \begin{align*} \mathrm{Spec}(\mathbf{C}[x]) &= (0)\cup\bigcup_{a\in\mathbf{C}}(x-a),\\ \mathrm{Spec}(\mathbf{C}[x,y]) &= (0)\cup\bigcup_{a,b\in\mathbf{C}}(x-a,y-b)\cup\{p(x,y):\text{p\in\mathbf{C}[x,y] irreducible}\}\\ \mathrm{Spec}(\mathbf{Z}_p) &= (0)\cup(p) \end{align*} and $$\mathrm{Spec}(\mathbf{Z}[x])$$ is known and described in Spectrum of $\mathbb{Z}[x]$. Do we know what are the $$\mathrm{MSpec}$$ of the underlying pointed monoids of these four rings?