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Given a set $ X$ and a closure operator $ \text{cl}:2^X\to 2^X$ on $ X$ , if we define $ \psi:2^X\to 2^X$ s.t. $ \psi(Q)=\{q\in Q:q\in\text{cl}(Q\setminus\{q\})\}$ then is it true that $ \forall S\subseteq X(\psi(\psi(S))=\psi(S))$ (i.e. that $ \psi$ is idempotent) implies $ \text{cl}$ is the closure operator of some matroid? I know if $Read more

In this article Hyttinen and Kangas claim that universal abstract elementary classes which are categorical are essentially classes of vector spaces as soon as the model reaches a critical cardinality. The proof is grounded on the fact that they can equip the class with a closure operator which is Urbanik. I am not practical withRead more

Let $ R$ be a Noetherian normal domain, and let $ R^+$ denote the absolute integral closure of $ R$ , i.e. the directed union of all module-finite extension domains. I’m trying to understand what “separable” should mean for elements of $ R^+$ . It’s hard to say what “should mean” means, exactly, but let’sRead more

Simply stated, I’ve been trying for a long time to either find in the literature, or derive myself, a notion of path in Cech closure spaces, that specialises to paths in a topological space, and to graph-like paths in so-called “quasi-discrete closure spaces”. Let me recall the definitions: A closure space is a pair $Read more

While I’ve marked javascript and react as tags for this question, I believe this question is more about a generic design pattern issue, so, nor js or react skills are required to answer. In the below snippets you’ll find the same example being implemented in two (different) ways. While they both just work fine, IRead more

Apologies if this is not regarded as suitable for MO, but I have noticed other “soft” questions and feel this deserves some attention by the community, and did not know where else this could be asked. According to the Closure announcement of the Math2.0 Forum: It is with a little regret that I am announcingRead more