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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.

W. G.
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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

ImNotTheSaxMan
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  • I appreciate your honesty on this question. It's just that the definition here has it marked differently than what you you said (I mean you could be very well right). I know that isomorphism is what people really care about, but I am dealing with all these direct sums which all of the sudden use equal instead of isomorphism, and I just want an exact definition of two matroids being equal. I do appreciate your help! – W. G. Dec 03 '18 at 14:54
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    @W.G. I feel you on that. Can you provide a little context about the direct sums you are working with, and where the equals signs come up? – ImNotTheSaxMan Dec 03 '18 at 14:58
  • There are a lot of propositions that use equal here which depending on the situation might mean different things. I don't want to go off topic here and go into one specific example just because it might mean something different somewhere else, but I truly appreciate your help though to look at that! I just want to stick with the definition here. Thank you though, I appreciate it. – W. G. Dec 03 '18 at 15:11