On the sheaf representation of a $pq$-Baer $*$-semiring with involution

In the paper, a functional (sheaf) representation of a $pq$-Baer $*$-semiring with involution is obtained. For a $*$-semiring, the notions of central and central prime ideals are introduced. The set ${\rm Sp}\,S$ of all central prime ideals of a $pq$-Baer $*$-semiring with the Zariski topology becomes a zero-dimensional compact Hausdorff space. The sheaf $(\mathbb{L}(S), {\rm Sp}\,S)$ of $*$-semirings is constructed on ${\rm Sp}\,S$ as a basis space. It is proved that an arbitrary $pq$-Baer $*$-semiring is $*$-isomorphic to the $*$-semiring of all global sections of the sheaf $\mathbb{L}(S)$. Open questions are formulated.