Outer inverses: Characterization and applications

We characterize the elements with outer inverse in a semi-group S, and provide explicit expressions for the class of outer inverses b of an element a such that bS subset of yS and Sb subset of Sx, where x, y are any arbitrary elements of S. We apply this result to characterize pairs of outer inverses of given elements from an associative ring R, satisfying absorption laws extended for the outer inverses. We extend the result on right-left symmetry of aR circle plus bR = (a + b)R (Jain-Prasad, 1998) to the general case of an associative ring. We conjecture that 'given an outer inverse x of a regular element a in a semi-group S, there exists a reflexive generalized inverse y of a such that x <=(-) y' and prove the conjecture when S is an associative ring. (C) 2016 Elsevier Inc. All rights reserved.