# filter bases

## Chain-meet-closed sets ★★

Author(s): Porton

Let $\mathfrak{A}$ is a complete lattice. I will call a \emph{filter base} a nonempty subset $T$ of $\mathfrak{A}$ such that $\forall a,b\in T\exists c\in T: (c\le a\wedge c\le b)$.

\begin{definition} A subset $S$ of a complete lattice $\mathfrak{A}$ is \emph{chain-meet-closed} iff for every non-empty \Def[chain]{Total_order#Chains} $T\in\mathscr{P}S$ we have $\bigcap T\in S$. \end{definition}

\begin{conjecture} A subset $S$ of a complete lattice $\mathfrak{A}$ is chain-meet-closed iff for every filter base $T\in\mathscr{P}S$ we have $\bigcap T\in S$. \end{conjecture}