Definition:Filter in Ordered Set

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Definition

Let $\struct {S, \preceq}$ be a preordered set.

Let $F$ be a subset of $S$.

$F$ is a filter in $\struct {S, \preceq}$ if and only if:

$F$ is non-empty, filtered and upper.

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 WAYBEL_0:mode 4