If Compact Between then Way Below
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Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.
Let $x, k, y \in S$ such that:
- $x \preceq k$ and $k \preceq y$ and $k \in \map K L$
where $\map K L$ denotes the compact subset of $L$.
Then $x \ll y$
where $\ll$ denotes the way below relation.
Proof
By definition of compact subset:
- $k$ is compact.
By definition of compact:
- $k \ll k$
Thus by Preceding and Way Below implies Way Below:
- $x \ll y$
$\blacksquare$
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_8:1