On the representation of partially ordered sets (Q1360190)

A poset \(P\) is left meet-distributive (LMD) if for all \(x,y,z\) of \(P\), if \(x\wedge(y\vee z)\) exists then also \((x\wedge y)\vee(x\wedge z)\) exists and they are equal. \(P\) is supremum-dense if each \(b\) of \(P\) is the join of all completely join-irreducible elements below \(b\). An isomorphism \(f\) from \((P,\leq)\) onto \((Q,\subseteq)\) is called a neatest representation if for any \(S\subseteq P\) we have \[ f\bigl(\bigwedge \{a_i;\;i\in I\}\bigr)= \bigcap\bigl\{f(a_i);\;i\in I\bigr\} \] \[ f\bigl(\bigvee \{a_i;\;i\in I\}\bigr)= \bigcup\bigl\{f(a_i);\;i\in I\bigr\}. \] The result: Let \(P\) be an LMD poset where every nonempty chain has an infimum. Them \(P\) is supremum-dense iff \(P\) has a neatest representation.