Let $L$ be a lattice and $S \subseteq L$ be such that $\bigvee S$ exists in $L$. Then $\bigvee S$ is called an irredundant join if $\bigvee(S\setminus\{s\}) \neq \bigvee S$ for all $s \in S$.
I proved that if $\bigvee S$ is an irredundant join, then $S$ is an anti-chain.
Now, the question that I ask help is the following.
Let $L$ be a lattice such that ${\downarrow} a$ has no infinite chains, for $a \in L$ and let $J(L)$ be the set of join-irreducible elements of $L$.
Show that every element $a \in L$ has a representation as an irredundant join, $a = \bigvee S$, of a finite subset $S$ of $J(L)$.
I think I can answer my question now.
For each $a \in L$, it is $a = \bigvee J_a$, in that $J_a = \{ x \in J(L) : x \leq a \}$.
Since there is no infinite chains in ${\downarrow}a$, there is $F \subseteq J_a$ finite with $\bigvee F = a$, and can suppose $F$ is a anti-chain.
If $\bigvee(F-\{x\}) = a$, for some $x \in F$, eliminate $x$ from $F$. We can proceed like that because $F$ is finite. When there is no more $x$ like above, $F$ is irredundant.