Sheaf representations and characterizations of $pq$-Baer semirings with involution

N. S. Protasov, V. V. Chermnykh

Syktyvkar State University

Abstract: 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.

UDC: 512.55

MSC: 16Y60

Received: 22.12.2024
Revised: 23.01.2025
Accepted: 27.01.2025

DOI: 10.21538/0134-4889-2025-31-1-166-174

Bibliographic databases:

•