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

Now as a part my research of filters I attempt to solve this problem (the problem seems not very difficult and I hope to prove it today or tomorrow, however who knows how difficult it may be):

**Definition** A subset $latex S$ of a complete lattice $latex \mathfrak{A}$ is *chain-meet-closed* iff for every non-empty chain $latex T\in\mathscr{P}S$ we have $latex \bigcap T\in S$.

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

## 1 thought on “Chain-meet-closed sets on complete lattices”