Locally Integral Involutive PO-Semigroups

Abstract

We show that every locally integral involutive partially ordered semigroup (ipo-semigroup) $\mathbf{A}=(A,\le ,\cdot ,\sim ,-)$, and in particular every locally integral involutive semiring, decomposes in a unique way into a family $\{{\mathbf{A}}_p:p\in A^{+}\}$ of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are unital semirings. Moreover, we show that there is a family of monoid homomorphisms $\Phi =\{{\phi }_{pq}:{\mathbf{A}}_p\to {\mathbf{A}}_q:p\le q\}$, indexed over the positive cone $(A^{+},\le )$, so that the structure of $\mathbf{A}$ can be recovered as a glueing ${\int }_{\Phi }{\mathbf{A}}_p$ 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 $\{{\mathbf{A}}_p:p\in D\}$, indexed over a join-semilattice $(D,\vee )$ along a family of monoid homomorphisms $\Phi$ is an ipo-semigroup.