Definition:Meet-Continuous Lattice
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Definition
Let $\left({S, \preceq}\right)$ be a meet semilattice.
Then $\left({S, \preceq}\right)$ is meet-continuous if and only if
- $\left({S, \preceq}\right)$ is up-complete and
- (MC): for every an element $x \in S$ and a directed subset $D$ of $S$: $x \wedge \sup D = \sup \left\{ {x \wedge d: d \in D}\right\}$
Also known as
Some sources prefer not to hyphenate: meet continuous lattice.
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 WEYBEL_2:def 7