Operators consistent in regularity

If S(X) is an arbitrary subset of L(X) (where L(X) is the set of all bounded operators on a Banach space X), then we say that B is an element of L(X) is S-consistent, or consistent in S(X), provided that for all A is an element of L(X) the following holds: AB is an element of S(X) if and only if BA is an element of S(X). It is convenient to take that S(X) is close to the set of all invertible operators on X, or that S(X) contains regular operators. Here "regular" means that S(X) is equal to the set of invertible, left (right) invertible, Fredholm, left (right) Fredholm, Weyl, or Browder operators on X. In this article we completely describe operators consistent in the previous regularities.