Join-continuity applied to chains?

Question

In Lemma 2.3 of https://www.ams.org/journals/tran/1960-096-01/S0002-9947-1960-0118690-9/S0002-9947-1960-0118690-9.pdf, they state that every algebraic lattice $L$ is join continuous, i.e. $a \wedge \bigvee B=\bigvee (a \wedge B)$ for every lattice-ideal $B$ of $L$ and $a \in L$. A little bit further in the same paper, namely in Lemma 3.4, they use the join-continuity to say that $a \wedge \bigvee(X \vee c)=\bigvee (a \wedge (X \vee c))$, where $X$ is a chain (i.e. totally ordered subset). Why does this hold? Chains are in general not lattice-ideals.

Answer 1

The authors do not cite Lemma 2.3 in their proof of Lemma 3.4, so we should not assume that their argument has any gap. Lemma 2.3 is true as written, and Lemma 3.4 is true as written. It's just that we can't apply Lemma 2.3, as written, where it might be useful in Lemma 3.4. Instead, one can do one of the following two things.

First, the proof of Lemma 2.3 works even if you replace the lattice ideal $B$ with any up-directed set $B'$. (Here $B'$ is up-directed if whenever $a, b\in B'$ there exists $c\in B'$ such that $a\leq c$ and $b\leq c$.) Now, chains are up-directed, so one can use this stronger version of Lemma 2.3 in the proof of Lemma 3.4.

Alternatively, you can leave Lemma 2.3 as it is. Then, in Lemma 3.4, replace $X$ with the lattice ideal it generates: $X' = \{a\in L\;|\;(\exists x\in X)(a\leq x)\}$. Then $X'$ is a lattice ideal, and it can be used in place of $X$ in the proof of Lemma 3.4.