On the Pierce Stalks of Semirings With Proper Involution

Abstract

In this paper, we study the relationship between local and global conditions for proper and semiproper involutions in *-semirings. An involution is said to be proper if aa* = 0 implies a = 0. A semiproper involution is defined by a weaker condition: aSa*= 0 implies a = 0. It is established that in Rickart (pq-Baer) *-semirings any involution is proper (semiproper). It is proved that in the presence of a semiproper involution, the set of all central complemented idempotents coincides with the set of all central projections, which can be used in the study of the Pierce sheaf of *-semirings. The main result of the paper is the proof of the following results: 1) an involution in a *-semiring S is proper if and only if it is proper in all stalks of the Pierce sheaf of *-semiring S; 2) an involution in a *-semiring S whose Boolean algebra of central projections is finite is semiproper if and only if it is semiproper in all stalks of the Pierce sheaf of *-semiring S.