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(1) I have an element or object belongs to a structure under some condition. If the evaluation result of condition changes, then the element belongs to another structure. Is there a known mathematical structure to abstract the dynamic changes ?

(2) I have an element or object belongs to multiple structures at the same time. Is there a known mathematical structure to abstract the multiplicities ?

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    $\begingroup$ This might be interesting (to me) but I don't quite understand what you're saying. Can you provide examples of (1) and (2)? $\endgroup$ – Ethan Bolker Nov 1 '17 at 16:59
  • $\begingroup$ Well, you can always look at structures-of-structures. For example, towards (2) if I have a group $G$ then I also get a partial order $S(G)$ of subgroups of $G$, and elements of $G$ will in general belong to multiple elements of $S(G)$. Does this seem reasonable to you, or are you looking for something else? Meanwhile I don't really understand (1) at all (and I'm not totally sure I understand (2)), can you clarify? $\endgroup$ – Noah Schweber Nov 1 '17 at 17:00
  • $\begingroup$ This sounds really vague. What are you trying to do with these objects? For example anytime you have a lie group, it has the structure of both a manifold and a group. A topological ring is a topological space, a ring (and the additive abelian group underneath it). $\endgroup$ – RKD Nov 1 '17 at 17:26