The Generalized MP Inverse Of Category And Idempotency Projections

Some properties of the generalized Moore-Penrose inverse of morphisms and idem-potency projection under seminorms are studied in this paper.The notion of epic-monic generalized factorization of morphisms in the preadditivecategory are defined ,and existential conditions and expressions for the Moore-Penroseinverses with epic-monic generalized factorization are gived;Eight kinds of the charac-terizations on the reverse law are obtained;Some new necessary and su?cient conditionsof the existence and expressions of generalized Moore-Penrose inverses for morphismsare given .A complete solution is established to the problem of characterizing all situa-tions,where a linear combination of two di?erent projections and three di?erent projec-tions which satisfy some special condition are also an projection under seminorms.