A Counting of the Minimal Realizations of the Posets of Dimension Two

Abstract

The posets of dimension $2$ are those posets whose minimal realizations have two elements, that is, which may be obtained as the intersection of two of their linear extensions. Gallai’s decomposition of a poset allows for a simple formula to count the number of the distinct minimal realizations of the posets of dimension $2$. As an easy consequence, the characterization of M. El-Zahar and of N.W. Sauer of the posets of dimension $2$, with an unique minimal realization, is obtained.