A class of outer generalized inverses

In any *-semigroup or semigroup S, it is shown that the Moore-Penrose inverse y = a(dagger), the author's pseudo-inverse y = a', Chipman's weighted inverse and the Bott-Duffin inverse are all special cases of the more general class of "(b, c)-inverses" y is an element of S satisfying y is an element of (bSy) boolean AND (ySc), yab = b and cay = c. These (b, c)-inverses always satisfy yay = y, are always unique when they exist, and exist if and only if b is an element of Scab and c is an element of cabS, in which case, under the partial order M of Mitsch, y is also the unique M-greatest element of the set X-a = X-a,X- b,X- c = {x : x is an element of S, xax = x and x is an element of (bSx) boolean AND (xSc)} and the unique M-least element of Z(a) = Z(a,b,c) = {z : z is an element of S. zaz = z, zab = b and caz = c}. The above all holds in arbitrary semigroups S. hence in particular in any associative ring R. For any complex n x n matrices a, b, c, an efficient uniform procedure is given to compute the (b, c)-inverse of a whenever it exists. In the ring case, a is an element of R is called "weakly invertible" if there exist b, c is an element of R satisfying 1 - b is an element of (1-a)R, 1 - c is an element of R(1 - a) such that a has a (b, c)-inverse y satisfying ay = ya, and it is shown that a is weakly invertible if and only if a is strongly clean in the sense of Nicholson, i.e. a = u + e for some unit u and idempotent e with eu = ue. (C) 2011 Elsevier Inc. All rights reserved.