Given an infinite distributive lattice $L$, does $L$ contain a non-principal prime ideal $I$, or a non-principal prime filter $F$? ($I$ is said to be principal if there is $x\in L$ such that $I=\{y\in L: y\leq x\}$. Dual definition for filters.)


If the answer is "yes" to the question above, what if we replace "prime" by "maximal"?



Non-principal prime ideals in infinite distributive lattices
It’s not clear to me what exactly is the intended definition of prime ideals and filters in distributive lattices. Based on an analogy with other classes of structures, it seems to me that conceptually the best choice should be to make the definition correspond to subdirectly irreducible factors in subdirect products, that is, to surjective homomorphisms to the 2-element lattice. This means that prime ideals and filters are required to be both nonempty and proper.


In that case, $\omega$ with its usual linear order is an infinite lattice in which all prime ideals and prime filters (indeed, all nonempty filters and nonempty proper ideals) are principal.


Different choices in the definition may yield different answers; in particular the empty ideal/filter is always nonprincipal if allowed.


The answer for maximal ideals/filters is unequivocally “no”: for example, one can take any infinite distributive lattice $L$ with a top element $\top$ and a bottom element $\bot$ such that $L\smallsetminus\{\top,\bot\}$ has a top element $t$ and a bottom element $b$. Then the only maximal ideal in $L$ is the one generated by $t$, and dually for filters.


