I am reading Davey & Priestley's Introduction to lattices and order, and there they proved that if a lattice satisfies the ascending chain condition then for every non empty subset $A$ of $P$ there exists a finite $F$ such that $\bigvee A = \bigvee F$. I was wondering if this result also holds for complete lattices.
In a complete lattice, is every join of arbitrary elements equal to a join of a finite number of elements
lattice-ordersorder-theory
Best Answer
No, certainly not. Let $P$ be the lattice of subsets of an infinite set, say $\mathbb{N}$, and let $A$ be the family of singletons $\{ n \}, n \in \mathbb{N}$. Then their union is $\mathbb{N}$ but the union of any finite subfamily of $A$ is finite. Note that $P$ has a strictly ascending chain $\emptyset \subsetneq \{ 1 \} \subsetneq \{ 1, 2 \} \subsetneq \{ 1, 2, 3 \} \dots $.