Filtered iff Finite Subsets have Lower Bounds
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Theorem
Let $\struct {S, \precsim}$ be a preordered set.
Let $H$ be a non-empty subset of $S$.
Then $H$ is filtered if and only if:
Proof
This follows by mutatis mutandis of the proof of Directed iff Finite Subsets have Upper Bounds.
$\blacksquare$
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_0:2