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.
Keywords: semiring with involution, $pq$-Baer $*$-semiring, sheaf representation
Received October 27, 2023
Revised November 21, 2023
Accepted December 4, 2023
Nikita Sergeevich Protasov, doctoral student, Pitirim Sorokin Syktyvkar State University, Syktyvkar, 167001 Russia, e-mail: protasovnekit@gmail.com
Vasiliy Vladimirovich Chermnykh, Dr. Phys.-Math. Sci., Prof., Pitirim Sorokin Syktyvkar State University, Syktyvkar, 167001 Russia, e-mail: vv146@mail.ru
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Cite this article as: N.S. Protasov, V.V. Chermnykh. On the sheaf representation of a $pq$-Baer $*$-semiring with involution. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 190–202.
