Universal Factorization And Generalized Moore-Penrose Inverses Of Morphisms

The generalized inverse of the Fredholm integral operators was given by Predholm,and the solution of the integral oprators equation was obtained.The simpler characterization of generalized inverse was given with four matrices equation by R.Penrose.Since then,Many mathematicians have been engaged in studying the generalized inverse of matrices.The aim of this paper is to study the Generalized Moore-Penrose Inverse and Generalized Moore-Penrose Inverse with epic-monic universal factorization of morphisms. The main results are liste in the following:(l)In the first chapter ,we study the generalized Moore-Penrose inverse with epic-monic factorization, The main results are proved as follows:Theorem 1.5:Let f : X → Y be a morphism of a category C,f = f₁f₂ is an epic-monic factorization of f,then the following statements are equivalent : 1)f has a generalized Moore-Perose inverse with respect to h, k;2)f₁^*hf₁,f₂k^-1f₂^* are 1eft (or right) invertible;3)f₁, f₂ has a generalized Moore-Perose inverse with respect to h, k;4) f₁*hf₁,f₂k^-1f₂^* are epic(or monic) moiphisms,f₁*hf₁,f₂k^-1f₂^* have generalized Moore-Perose inverse with respect to h, k;5)f^*hf₁ is an epic morphism f₂k^-1 f^* is an monic morphism, f₁*hf₁,f₂k^-1f₂^* have generalized Moore-Perose inverse with respect to h, k;6)f₁*hf₁,f₂k^-1f₂^* are epic(or monic) morphisms,f₁*hf₁,f₂k^-1f₂^* have generalized Moore-Perose inverse with respect to h, k.(2)In the second chapter,the author introduced generalized factorization and discuss necessary and sufficient conditions for existence and expression of generalized Moore-Penrose inverse with generalized factorization,we generalize the corresponding results of morphisms with universal factorization , The main results are proved as follows:Theorem 2.11:Let f = pgq be a generalized factorization , and g{1} ≠0,then the following statements axe equivalent :1)/ has a generalized Moor-Perose inverse;2)p$M{l,3} ± <^,and gqhJt{l,4} ^ ;3) There exists 01,0:2 such that axf*hpg = g = gqk¹f*ot2-(3) In the third chapter we dinfined epic-monic universal factorization of morphisms,and give existential conditions and expressions for the epic-monic universal factorization of morphisms.we prove the following:Theorem 3.5:Let / € Hom(X,X),f pgq be a morphism of a category has a involution * , There exist morphismsp' e Hom(Y, X), q' e Hom(X, Y), such that p'pg = g = gqq',then the following statements are equivalent :3 There exist a ï¿¡ Hom(X, X) , such that pg = crf*pg, gq = gqf*&;4)pg is *— letf cancelable , and (pg)*pg is a regular , gq is *—right cancelable , and gq(gq)* is a regular ?Theorem 3.6:Let / € Hom(X, X), f = pgq be a morphism of a category has a involution * , There exist morphisms;/ G Hom(Y, X),q' € Hom(X, Y), such that p'pg = g = gqq',and pg is an epic morphism,<7<7 is an monic morphism,then the following statements are equivalent :l)/has a Moore-Perose inverse with respect to *;2)f*pg is left invertible,gg/* is right invertible;3)(pg)*pg,gq(gq)* are both invertible ?...