Let pP, ĺ1q and pQ, ĺ2q be two posets. Their product poset is defined as the product set P ˆQ, togetherwith a partial order relation ĺ, where we say that:pp1, q1q ĺ pp2, q2q if and only if p1 ĺ1 p2 and q1 ĺ2 q2.For the purposes of this problem, let P and Q be the divisor posets of 8 and 9 respectively, with thestandard (divisibility) order relations in each case. That is, P consists of all the divisors of 8 and Qof all the divisors of 9.(a) How many elements does the product poset P ˆ Q have? Justify briefly.(b) Draw the Hasse diagram of P ˆ Q, labelling each node.
Question
Let pP, ĺ1q and pQ, ĺ2q be two posets. Their product poset is defined as the product set P ˆQ, togetherwith a partial order relation ĺ, where we say that:pp1, q1q ĺ pp2, q2q if and only if p1 ĺ1 p2 and q1 ĺ2 q2.For the purposes of this problem, let P and Q be the divisor posets of 8 and 9 respectively, with thestandard (divisibility) order relations in each case. That is, P consists of all the divisors of 8 and Qof all the divisors of 9.(a) How many elements does the product poset P ˆ Q have? Justify briefly.(b) Draw the Hasse diagram of P ˆ Q, labelling each node.
Solution
(a) The divisors of 8 are 1, 2, 4, and 8. The divisors of 9 are 1, 3, and 9. Therefore, the product poset P ˆ Q will have 4 * 3 = 12 elements. This is because each element in P can be paired with each element in Q, and there are 4 elements in P and 3 elements in Q.
(b) The Hasse diagram of P ˆ Q will have 12 nodes, each representing a pair (p, q) where p is a divisor of 8 and q is a divisor of 9. The pairs are (1,1), (1,3), (1,9), (2,1), (2,3), (2,9), (4,1), (4,3), (4,9), (8,1), (8,3), (8,9).
The partial order relation is defined as (p1, q1) ≤ (p2, q2) if and only if p1 divides p2 and q1 divides q2. Therefore, the Hasse diagram will have an edge from (p1, q1) to (p2, q2) if and only if p1 divides p2 and q1 divides q2.
For example, there will be an edge from (1,1) to all other nodes, because 1 divides all numbers. There will also be an edge from (2,1) to (4,1), (8,1), (2,3), (4,3), (8,3), (2,9), (4,9), (8,9) because 2 divides 4 and 8, and 1 divides all numbers.
The rest of the edges can be drawn in a similar manner. The resulting Hasse diagram will be a visual representation of the product poset P ˆ Q.
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