  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
$\Phi = \{\varphi_{pq}: \mathbf A_p\to \mathbf A_q : p\le q\}$, 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
P{\l}onka sum of any family of integral ipo-monoids $\{\mathbf A_p : p\in D\}$,
indexed over a join-semilattice $(D,\lor)$ along a family of monoid
homomorphisms $\Phi$ is an ipo-semigroup.
