Some remarks on normalized matching

A new class of LYM orders is obtained, and several results about general LYM orders are proved. (1) Let A₁ ? A₂ ? … ? A_r be a chain of subsets of n] = {l,…,n}. Let 〈a_i〉 and 〈b_i〉 be two nondecreasing sequences with a_i ? b_i for l ? i ? r. Then {X ? n]: a_i ? | ∩ A_i|? b_i}, ordered by inclusion, is a poset having the LYM property. (2) The smallest regular covering of an LYM order has M(P) chains, where M(P) is the least common multiple of the rank sizes. (3) Every LYM order has a smallest regular covering with at most || ? h(P) classes of distinct chains, where h(P) is the height of P. To obtain (3), we discuss “minimal sets” of covering relations between two adjacent levels of an LYM-order.