Let $(A,\geq)$ be a partially ordered set such that
- there exists the join $\bigvee A$, i.e. $a\in A$ such that $a\geq b$ for any $b\in A$;
- for any pair $(b,c)\in A\times A$ there exists the meet $b \wedge c\in A$, i.e. $d\in A$ such that $d\leq b,c$ and such that $d\geq e$ for any other $e\in A$ satisfying $e\leq b,c$.
Is it possible to construct a bijection $f: \eta\to A$ for $\eta$ ordinal number such that for any $\alpha<\beta\leq \eta$ there exists $\gamma\leq\beta$ such that $$ f(\alpha)\wedge f(\beta)= f(\gamma)\, \,? $$ May you suggest some references for my problem? I am new in the field of set theory.