If $\kappa$ is an infinite cardinal, is there a lattice $L$ of cardinality $\kappa$ such that $L$ contains no prime ideals?
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Yes - set $L:=\kappa+2$ and endow it with the following ordering:
- $\kappa < \alpha$ for all $\alpha \in \kappa$;
- $\kappa+1 > \alpha$ for all $\alpha \in \kappa$.
(Essentially this is an infinite version of the non-distributive lattice $M_3$ with a $\kappa$-antichain.)
The only proper ideal is the singleton consisting of the bottom element. But this ideal is not prime as the meet of any two members of $\kappa$ equals the bottom element.
answered Jul 14, 2015 at 7:22
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1$\begingroup$ The principal ideals generated by any $\alpha\in\kappa$ are all prime. $\endgroup$ Commented Jul 14, 2015 at 13:33
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$\begingroup$ Ah, sorry: they are meet-irreducible, but not prime. However, they are most definitely proper ideals, so they need to be dealt with in the answer in one way or another. $\endgroup$ Commented Jul 14, 2015 at 13:39