Every non-selfadjoint operator algebra $A$ generates a C*-algebra, but isomorphic copies of $A$ can generate many non-isomorphic C*-algebras, and we call these the C*-covers of $A$. A celebrated result--first proved by Hamana, is that a unique minimum C*-cover for any $A$ exists, called the C*-envelope. The C*-envelope is intrinsic to $A$, but non-isomorphic operator algebras $A$ and $B$ can share the same C*-envelope. If we instead ask that $A$ and $B$ share ALL the same C*-covers, must $A$ and $B$ be isomorphic?

The C*-covers of an operator algebra form a complete lattice, but little more is known regarding the structure of this lattice. There are multiple natural senses in which two operator algebras may have "the same" lattice of C*-covers, and we will discuss how these different senses remember different information about the operator algebras involved. This is joint work with Dr. Christopher Ramsey.