# filter bases

## Chain-meet-closed sets ★★

Author(s): Porton

Let is a complete lattice. I will call a *filter base* a nonempty subset of such that .

**Definition**A subset of a complete lattice is

*chain-meet-closed*iff for every non-empty chain we have .

**Conjecture**A subset of a complete lattice is chain-meet-closed iff for every filter base we have .

Keywords: chain; complete lattice; filter bases; filters; linear order; total order