Names of definitions in frames/locales/topologies

Question

Background definitions.

• A frame $(\mathbb X,\leq)$ is a partially ordered set with arbitrary joins (least upper bounds) and finite meets (greatest lower bounds), such that finite meets distribute over arbitrary joins.
• A filter $F\subseteq \mathbb X$ is a nonempty up-closed proper subset of $X$ that is closed under finite meets.

Example: A standard example of a frame is the lattice of open sets of a topology.

Consider the following definitions, for $F\subseteq\mathbb X$ (we are most interested in this definition when $F$ is a filter, but the definition does not require it) and $x,x'\in\mathbb X$:

1. $x\ast x' \Leftrightarrow x\wedge x'\neq\bot_{\mathbb X}$ (intuitively: $x\ast x'$ when $x$ and $x'$ intersect).
2. $x\ast F \Leftrightarrow \forall x'{\in}F. x\ast x'$ (intuitively: $x\ast F$ when $x$ intersects every element in $F$).
3. $F^c = \{ x\in\mathbb X \mid x\not\in F\}$
4. $F^\ast = \{ x\in\mathbb X \mid x\ast F\}$.
5. $F^{\ast c} = (F^\ast)^c = \{ x\in\mathbb X \mid \neg (x\ast F)\}$.

Example: Suppose $(\mathbb X,\leq)$ is the frame of open sets of a topology $(\mathbb P,\mathit{Opens})$, and $p\in\mathbb P$, and write $\mathit{nbhd}(p)=\{O\in\mathit{Opens} \mid p\in O\}$ for the neighbourhood filter. Then $\mathit{nbhd}(p)^c = \mathbb P \setminus |p|$.

My questions are:

1. What are the standard names, if any, for $F^c$, $F^\ast$, and $F^{\ast c}$?
2. Is there a canonical reference for where these are studied?

Thank you.