Definition: Matroids being Equal

Question

I've been trying to find a definition of two matroids being equal. I would have thought two matroids are equal iff their ground sets and independent sets are equal. However, online I found out that two matroids are equal iff their ground sets have the same size/cardinality and their independent sets are equal (the definition here says indexed bases). This is the only definition I can find for two matroids being equal, and I just wanted to verify that other people use this definition. It's not defined in the index of Oxley's book as well as the other book I use which I predominantly use which is why I just wanted to double check to see if this is the definition everyone uses.

Answer 1

I'm fairly sure that two matroids cannot be equal in the strictest sense if their ground sets are different. However, the whole point of matroids are that the particular elements of the ground set don't actually play a role; we don't care about matroid equality, we care about matroid isomorphism.

I think that this distinction between equality and isomorphism is best understood in the context of matroid intersections. If the ground sets are different but the matroids are isomorphic you can't perform a meaningful intersection. See the link below.

https://theory.stanford.edu/~jvondrak/CS369P/lec10.pdf