Does anyone know an example of a bounded domain, not L-domain, in which every principal ideal is a join semilattice

Question

I'm looking for an example of a bounded domain P in which every principal ideal is a Join semilattice but P isn't a L-domain. I have doubts If this kind of poset existe.

Answer 1

You're right: Every finite join-semilattice is a complete lattice. [Note: For me, being a join-semilattice includes having a bottom element $\bot$ (the empty join). It's not necessary to assume the existence of $\top$.]

Indeed, if $S$ is a finite join-semilattice, then $S$ has a top element $\top = \bigvee_{s\in S} s$. And for any $a,b\in S$, $a$ and $b$ have a meet $a\land b = \bigvee_{s\in S_{a,b}} s$, where $S_{a,b} = \{s\in S\mid s\leq a \text{ and }s\leq b\}$.

From these definitions, it's not hard to check that $(S,\bot,\top,\vee,\wedge)$ is a lattice. And any finite lattice is complete, because arbitrary joins are finite joins and arbitrary meets are finite meets.