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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

Set of lower bounds in poset is defined like $ A^l = \{ x \in P : \forall a \in A . x \le a \} = \bigcap_{a \in A} \{ x \in P : x \le a \}$. Is there in literature a name for union $ \bigcup_{a \in …