The study of π(K), the semiring of matrices which leave invariant the proper cone K, is continued in this paper. The definitions of Baer rings and Baer ∗-rings are extended to semirings, at least in the case of π(K). Results that relate the concepts of right (left) annihilators, right (left) facial ideals, and p-exposed or o.p.-exposed faces are given. Proper cones K for which π(K) is a right (left) Baer semiring or a Baer ∗-semiring are characterized. In particular, it is shown that π(K) is a Baer ∗-semiring if and only if K is a perfect cone. The set of projections in π(K) is investigated, and its lattice structure is analyzed. The concepts of equivalence of idempotents and ∗-equivalence of projections in π(K) are introduced and examined.
Linear Algebra and Its Applications 256, pp.165-183