Reverse order laws for -generalized inverses in -algebras

In this paper, we consider the reverse order laws for { 1 , 3 , 4 } -generalized inverses in C ∗ algebras. We present purely algebraic necessary and sufficient conditions for b { 1 , 3 , 4 } ⋅ a { 1 , 3 , 4 } ⊆ ( a b ) { 1 , 3 , 4 } and ( a b ) { 1 , 3 , 4 } = b { 1 , 3 , 4 } ⋅ a { 1 , 3 , 4 } . Also, we prove that ( a b ) { 1 , 3 , 4 } ⊆ b { 1 , 3 , 4 } ⋅ a { 1 , 3 , 4 } is actually equivalent to ( a b ) { 1 , 3 , 4 } = b { 1 , 3 , 4 } ⋅ a { 1 , 3 , 4 } .