Weighted (b,c)-inverses in categories and semigroups

In [Linear Algebra Appl. 436 (2012), 1909-1923], the author introduced, for any semigroup S and any the (b, c)-invertibility of a as meaning the existence of some (b, c)-inverse such that yab = b and cay = c (and also introduced two other weaker version for rings), which generalizes both the Moore-Penrose inverse and the author's pseudo-inverse for square matrices A. The present article further generalizes (b, c)-inverses in two directions, first so as to apply to morphisms in arbitrary categories (rather than only to elements of semigroups) and secondly so as to yield weighted versions of (b, c)-inverses by simultaneously generalizing J. S. Chipman's weighted version of and R. E. Cline and T. N. E. Greville's weighted version of To enable some of these developments, H. Mitsch's well-known partial order in semigroups is extended to obtain a version applicable to morphisms in arbitrary categories.