Semigraphoids are combinatorial structures arising from models of probabilistic conditional independence. We investigate the classification of semigraphoids on n elements. As a motivating example, we present a complete classification on three elements. By studying the construction of semigraphoids from generating sets, we establish two quite different classes of maximal semigraphoids. For every k, the semigraphoid generated by the non-k-atoms is maximal, and for every k the direct sum of a maximal semigraphoid on k elements and the complete model on n -- k elements is maximal. The semigraphoids on four elements are computed and their maximal semigraphoids classified highlighting these results
