Consider the semiring $$\mathbb{N}[H,H^{-1}]/(H^p+H^q = H^{p+q}+1)_{p,q \in \mathbb{Z}}.$$ Is it finitely presentable? Is there any simplification of the relations (except for $p \geq q \geq 0$)?

*Topological background.* Whereas the semiring of vector bundles on the scheme $\mathbb{P}^1_{\mathbb{C}}$ is just $\mathbb{N}[H,H^{-1}]$ (old result by Dedekind-Weber, 1892, which has been rediscovered many times), the semiring of vector bundles on the topological space $\mathbb{C}\mathbb{P}^1$ seems to be the semiring above, where $H$ represents the tautological bundle. The relations follow from Example 1.13 in Hatcher's VBKT. The reason is that $\begin{pmatrix} z^p & 0 \\ 0 & z^q \end{pmatrix}$ is homotopic to $\begin{pmatrix} z^{p+q} & 0 \\ 0 & 1 \end{pmatrix}$ in $\mathrm{Map}(S^1,\mathrm{GL}_2(\mathbb{R}))$. The $K$-theory ring is just $K(\mathbb{C}\mathbb{P}^1)=\mathbb{Z}[H]/(H^2+1=2H)$.