3
votes
1answer
301 views

Cardinality of the set of ultrafilters on an infinite Boolean algebra

Let $\mathfrak B$ be a Boolean algebra with an infinite power $\kappa$. My question is how many ultrafilters does it have? $\kappa$ or $2^\kappa$? Or even smaller?
4
votes
1answer
305 views

Is a chain-complete lattice a complete lattice without the axiom of choice?

This question is inspired by this question. Consider the following result: Let $(L,\leq)$ be a chain-complete lattice. Then $(L,\leq)$ is a complete lattice. Can this result be proven without ...
4
votes
1answer
192 views

Question about Boolean algebra and ultrafilters

In the following $B$ denotes a Boolean algebra and $\bar{x}$ is the complement of $x$. In my notes there is the following theorem: If $U \subset B$ is an ultrafilter on $B$ then for every $x \in B$ ...
3
votes
1answer
632 views

Complete lattices and sublattices – which requirement is more stringent?

I'm studying from Michael Carter's "Foundations", and on page 29 he makes the comment, "Note that the requirement of being a sublattice is more stringent than being a complete lattice in its own ...