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user 10231
Questions of the kind "What's the name for a X that satisfies property Y?"
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Name for union of upsets/downsets
Introduction to Lattices and Order by B. A. Davey and H. A. Priestly calls this $\mathord{\downarrow}A$ or the downset of $A$ and also uses $\mathord{\downarrow} a$ for $\{x \in P : x \le a\}$
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