Let $\mathcal{I}$ be an ideal of a poset $(P,\leq)$, i.e. $\mathcal{I}$ non-empty, and $$ \forall x,y\in P: x\leq y\text{ and }y\in\mathcal{I}\implies x\in\mathcal{I}\\ \forall x,y\in \mathcal{I}\ \exists z\in\mathcal{I}:\ x,y\leq z $$ Suppose that $\mathcal{I}$ has the property that for all $c\in P$, there exists $x\in\mathcal{I}$ with $x\leq c$. Has this property a name? Has it been studied? Does it come up in other, maybe more specific situations (e.g. when $\mathcal{I}$ is a set-theoretical ideal)?
Context: We say that a set $\mathcal{A}\subseteq\mathcal{P}(\mathbb{N})$ forces convergence if and only if for every real sequence $a\in\mathbb{R}^\mathbb{N}$ (and $a_{\infty}\in\mathbb{R}$) we have $$ a \text{ converges to } a_\infty \iff \forall A\in\mathcal{A}:\quad a|_{A}\text{ converges to } a_\infty $$ where to make things neat we adopt the convention that if $A\subset\mathbb{N}$ is finite, then $a|_A$ converges to every real number.
Let $\mathcal{I}(\mathcal{A}):=\{B\subseteq\mathbb{N}\ |\ \exists F\subset\mathbb{N}\text{ finite}, A_1,...,A_n\in\mathcal{A}:\ B\subseteq F\cup\bigcup_{i=1}^nA_i\}$ be the (set-theoretical) ideal generated by $\mathcal{A}$; we can see that $\mathcal{A}$ forces convergence if and only if $\mathcal{I}(\mathcal{A})$ forces convergence. Furthermore, knowing that a sequence converges to some limit if and only if every subsequence admits a subsequence converging to this same limit, we obtain $$ \mathcal{A}\text{ forces convergence }\iff \forall C\subseteq\mathbb{N}\ \exists A\in\mathcal{I}(\mathcal{A}):\ A\subseteq C, $$ i.e. $\mathcal{I}(\mathcal{A})$ has the property of the beginning of this question.