Let is a complete lattice. I will call a filter base a nonempty subset
of
such that
. I will call a chain (on
) a linearly ordered subset of
.
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 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
.
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