Dimension of the crown Skn

In 1941, Dushnik and Miller introduced the concept of the dimension of a poset (X, P) as the minimum number of linear extensions of P whose intersection is exactly P. Although Dilworth has given a formula for the dimension of distributive lattices, the general problem of determining the dimension of a poset is quite difficult. An equally difficult problem is to classify those posets which are dimension irreducible, i.e., those posets for which the removal of any point lowers the dimension. In this paper, we construct for each n≥3, k≥0, a poset, called a crown and denoted S^k_n, for which the dimension is given by the formula $\text{2?(n+k)}\text{(k+2)}$. Furthermore, for each t≥3, we show that there are infinitely many crowns which are irreducible and have dimension t. We then demonstrate a method of combining a collection of irreducible crowns to form an irreducible poset whose dimension is the sum of the crowns in the collection. Finally, we construct some infinite crowns possessing combinatorial properties similar to finite crowns.