Reverse order law for outer inverses and Moore-Penrose inverse in the context of star order

The reverse order law for outer inverses and the Moore-Penrose inverse is discussed in the context of associative rings. A class of pairs of outer inverses that satisfy reverse order law is determined. The notions of left-star and right-star orders have been extended to the case of arbitrary associative rings with involution and many of their interesting properties are explored. The distinct behavior of projectors in association with the star, right-star, and left-star partial orders led to several equivalent conditions for the reverse order law for the Moore-Penrose inverse.