Notation/terminology: Existence of a nonleast element which is less of any element of a set

Question

Let $A$ is a subset of a partial order $X$.

Are there any name and/or notation for the following predicate $P(A)$?

$P(A)$ iff there is a non-least element $x$ of $X$ which is a subelement of each element of $A$ (that is $\forall y\in A:x\leq y$)?

Answer 1

1. In set theory, we would probably say: "$A$ has a non-empty intersection."

2. So if you're working a complete lattice, you could say: "$A$ has a non-trivial meet."

3. Otherwise, I would say: "$A$ has a non-trivial lower bound."