Help to conceive a name _Filter_ $F$ is defined by the formula $$A\cap B\in F \Leftrightarrow A\in F\wedge B\in F.$$ _Ideal_ $F$ is defined by the formula $$A\cup B\in F \Leftrightarrow A\in F\wedge B\in F.$$ In my book I define _free star_ $F$ by the formula $$A\cup B\in F \Leftrightarrow A\in F\vee B\in F.$$ The last thing is $$A\cap B\in F \Leftrightarrow A\in F\vee B\in F,$$ it is yet unnamed. Please help to conceive a name for the $F$ from the last formula. Remark: There exists "natural" bijections between every two of the above defined four kinds of objects. They are essentially the same, that is.

Consider **co-ideal**.

This may be helpful in remembering the main property of this new item.

What I mean is that $F$ is ideal means $$\forall A,B : A \cup B \in F \equiv A \in F \land B \in F$$ and the dual of $\cup$ is $\cap$, and the dual of $\land$ is $\vee$, so `dualising' the formula for ideal yields the formula $$\forall A,B : A \cap B \in F \equiv A \in F \lor B \in F$$ which means $F$ is ``co-ideal'' :D

Hope that helps!