Let $P = (X, {\preceq})$ be a D1103: Partially ordered set.
The set of lower bounds of $E \subseteq X$ with respect to $P$ is the D11: Set
\begin{equation}
\mathsf{LB}_P(E) : = \{ m \in X \mid \forall \, x \in E : (m, x) \in {\preceq} \}
\end{equation}