For a poset \((P,\leq)\) it is known that there is an ordering \((Q,\leq)\) and an isomorphism \(f\) between \(P\) and \(Q\) with either \[ (*) \quad f\Bigl( \bigwedge_{i\in S} a_ i \Bigr)= \bigwedge_{i\in S} f(a_ i) \qquad \text{ or } \qquad (**) \quad f\Bigl( \bigvee_{i\in S} a_ i \Bigr)= \bigvee_{i\in S} f(a_ i) \] for any finite indexing set \(S\). In general you will not obtain \((*)\) and \((**)\). Now let \((P,\leq)\) be an ordering where every nonempty chain has an infimum and where every finite subset with an upper bound has an infimum. It is shown that in this case the existence of a mapping with \((*)\) and \((**)\) and \(S\) arbitrary is equivalent to the existence of a weakly dense set of completely join irreducible elements. Note that a subset \(D\subseteq P\) is weakly dense, if for any \(p,q\in P\) with \(p\not\leq q\) there is a \(d\in D\) with \(d \not\leq q\) and \(d\leq p\). Moreover \(x\in P\) is completely join irreducible, if whenever \(x\leq \bigvee_{i\in S} x_ i\) for an arbitrary indexing set \(S\), then \(x\leq x_ j\) for some \(j\in S\).