For A ≠ ∅, let (A, R) be a poset, and let ∅ ≠ B ⊆ A such that R' = (B × B) ⋂ R. If (B, R') is totally ordered, we call (B, R') a chain in (A, R). In the case where B is finite, we may order the elements of B by b1 R' b2 R' b3 R' ... R' bn-1 R' bn and say that the chain has length n. A chain (of length n) is called maximal if there is no element a ∈ A where a ∉ {b1, b2, b3,..., bn} and a R b1, bn R a, or bi R a R bi+1, for some 1 ≤ i ≤ n - 1.
(a) Find two chains of length 3 for the poset given by the Hasse diagram in Fig. 7.20. Find a maximal chain for this poset. How many such maximal chains does it have?
(b) For the poset given by the Hasse diagram in Fig. 7.18(d), find two maximal chains of different lengths. What is the length of a longest (maximal) chain for this poset?
(c) Let U = {1, 2, 3, 4} and A = P(U). For the poset (A, ⊆), find two maximal chains. How many such maximal chains are there for this poset?
(d) If U = {1, 2, 3, ..., n], how many maximal chains are there in the poset (P(U), ⊆)?