Examples of non-complete lattices with enough prime elements

Question

In this nice answer, Eric Wofsey shows that a lattice $L$ is isomorphic to the lattice of open sets of a topological space if, and only if, the following two conditions hold:

1. $L$ is complete: every subset $S\subseteq L$ has a least upper bound.

2. $L$ has enough prime elements (elements $p\in L\setminus\{1\}$ such that $a\wedge b\leq p$ implies $a\leq p$ or $b\leq p$): that is, if $a,b\in L$ and $a\not\leq b$, then there exist a prime element $p\in L$ such that $b\leq p$ and $a\not\leq p$.

Although I already know complete lattices without enough prime elements (here), I wonder about examples of non-complete lattices with enough prime elements. Since I am new to this pointless perspective, I don't know whether this question is trivial.

Anyway, I would like to ask for examples of lattices satisfying condition 2 but not condition 1, or references where I can find them.

Answer 1

For a very simple example, let $X$ be any infinite set and let $L$ be the lattice of subsets of $X$ that are either finite or cofinite. Then $L$ is not complete (for instance, if $A\subset X$ is infinite and coinfinite, the collection of finite subsets of $A$ has no join in $L$). However, $L$ has enough prime elements: the prime elements of $L$ are the complements of singletons, and if $a\not\leq b$ in $L$ then there is some $x\in a$ that is not in $b$ and the prime $p=X\setminus\{x\}$ satisfies $b\leq p$ and $a\not\leq p$.