Compact Subset is Bounded Below Join Semilattice
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Theorem
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.
Let $P = \left({K\left({L}\right), \precsim}\right)$ be an ordered subset of $L$,
where $K\left({L}\right)$ denotes the compact subset of $L$.
Then $P$ is a bounded below join semilattice.
Proof
- $\bot_L$ is a compact element,
where $\bot_L$ denotes the smallest element in $L$.
By definition of compact subset:
- $\bot_L \in K \left({L} \right)$
By definition of the smallest element:
- $\forall x \in K\left({L}\right): \bot_L \preceq x$
By definition of ordered subset:
- $\forall x \in K\left({L}\right): \bot_L \precsim x$
Thus by definition:
- $P$ is bounded below.
Thus by Compact Subset is Join Subsemilattice:
- $P$ is join semilattice.
$\blacksquare$
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 WAYBEL13:15