N.S. Protasov, V.V. Chermnykh. Sheaf representations and characterizations of pq-Baer semirings with involution
We study a semiring with involution in which the annihilator of an arbitrary principal right ideal is generated by the projection ($pq$-Baer $*$-semiring). For $*$-semirings, three sheaves are constructed, analogues of the Lambek, Pierce and Cornish sheaves. It is shown that for $pq$-Baer $*$-semirings these sheaves are isomorphic. This implies that an arbitrary $pq$-Baer $*$-semiring is $*$-isomorphic to the $*$-semirings of sections of these sheaves. A description of $pq$-Baer *-semirings without nilpotent elements and strongly Rickart $*$-semirings in terms of sections of sheaves is obtained. These results make it possible to clarify the structure of the elements of the indicated $*$-semirings.
Keywords: semiring with involution, $pq$-Baer $*$-semiring, strongly Rickart $*$-semiring, sheaves of $*$-semirings
Received December 22, 2024
Revised January 23, 2025
Accepted January 27, 2025
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., Chief Researcher, 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. Sheaf representations and characterizations of $pq$-Baer semirings with involution. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 1, pp. 166–174.
