Definition:Generator Set of Filter

Definition

Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice.

Let $F$ be a filter on $L$.

The generator set $G$ of $F$ is defined as follows:

$F = \left({\operatorname{fininfs}\left({G}\right)}\right)^\succeq$

where

$\operatorname{fininfs}\left({G}\right)$ denotes the finite infima set of $G$,

$G^\succeq$ denotes the upper closure of $G$.

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL12:def 3