The Structure of Locally Integral Involutive Po-monoids and Semirings

Abstract

We show that every locally integral involutive partially ordered monoid (ipo-monoid) $\textbf{A}= (A,\leqslant , \cdot , 1, {\sim },{-})$, and in particular every locally integral involutive semiring, decomposes in a unique way into a family $\{\textbf{A}_p : p\in A^+\}$ of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are semirings. Moreover, we show that there is a family of monoid homomorphisms $\Phi = \{\varphi _{pq}: \textbf{A}_p\rightarrow \textbf{A}_q : p\leqslant q\}$, indexed on the positive cone $(A^+,\leqslant )$, so that the structure of $\textbf{A}$ can be recovered as a glueing $\int _\Phi \textbf{A}_p$ of its integral components along $\Phi$. Reciprocally, we give necessary and sufficient conditions so that the Płonka sum of any family of integral ipo-monoids $\{\textbf{A}_p : p\in D\}$, indexed on a lower-bounded join-semilattice $(D,\vee ,1)$, along a family of monoid homomorphisms $\Phi$ is an ipo-monoid.

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