Let P be the partial order defined on the set {1, 2, 3, 4} as follows.

P = {(a, b) | a, b ∈ {1, 2, 3, 4}} with relation...

Question

Let P be the partial order defined on the set {1, 2, 3, 4} as follows.

P = {(a, b) | a, b ∈ {1, 2, 3, 4}} with relations: (1,1), (2,2), (3,3), (4,4), (1,3), (2,3), (2,4).

The number of total orders on {1, 2, 3, 4} that contain P is __________.

Accepted Answer

5 A total order (linear extension) of a partial order must keep all the existing comparisons intact and add a definite ranking for every incomparable pair. We count how many such total orderings exist by systematically choosing the next minimal element at each step. The key is to identify which elements are 'free' (minimal) at each stage and branch over all valid choices.