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I would like to trace the concepts "rank function" and "closure operator" back to some structures as primitive as possible.

For a set system $(E,F)$ which is an independence system or a greedoid, I have seen in literatures a rank function defined from $\mathcal P(E)$ to $\mathbb N$, as something like,

$$\forall A \in \mathcal P(E), \quad rank(A) := \max_{S \in F, S \subseteq A} |S|.$$

A "closure operator" (which is actually a semi-closure operator, not a closure operator yet in the sense for a closure system) is also defined on $\mathcal P(E)$, in terms of the predefined rank function, something like $$ \forall A \in \mathcal P(E), \quad \bar A := \{x \in E: rank (A) = rank(A\cup \{x\})\} $$

Since either an independence system or a greedoid is also an accessible system, I wonder if a rank function and a "closure operator" has been defined for a structure on an accessible system, or some more primitive structure than an independence system or a greedoid, or even an arbitrary set system?

Is such an "closure operator" always defined in terms of a predefined rank function?

Thanks.

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It is a bit unclear what you are trying to ask in this question. Perhaps it would be a good idea to include a couple definitions here and to ask more specific questions. –  Joseph Van Name Jun 23 at 0:00

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