A chain of length k in a poset is a finite sequence p1 p2 pk of length k, of strict inequalities in the poset. Show that the (i, j)th entry of (M M ) k counts the number of chains of length k that start at pi and end at pj .

Let (P, 3) be a poset, and fix a topological sorting of the elements of P. ) Let P be the "bow-tie" poset drawn earlier with a chosen topological sorting. Find by hand an element E Ap such that u* f = d. You may find it useful to move back and forth between the matrix representations of these functions. Let (P, 3) be a poset, and fix a topological sorting of the elements of P. ) Let P be the "bow-tie" poset drawn earlier with a chosen topological sorting. Find by hand an element E Ap such that u* f = d. You may find it useful to move back and forth between the matrix representations of these functions