如何证明格中的吸收律？含实例验证及一般化证明问询

To prove the absorption laws for any lattice (L) (a poset where every pair of elements has a supremum (\vee) and infimum (\wedge)), we’ll lean on the core definitions of these bounds and poset properties.

Proof of (a \wedge (a \vee b) = a) #

Let’s walk through this step by step:

1. Start with the supremum property: By definition, (a \vee b) is the least upper bound of (a) and (b). This means it’s greater than or equal to both elements, so (a \leq a \vee b) is always true.
2. Verify (a) is a lower bound: A lower bound of a set must be ≤ every element in the set. We know:
   • (a \leq a) (reflexivity of posets)
   • (a \leq a \vee b) (from step 1)
     So (a) qualifies as a lower bound for the set ({a, a \vee b}).
3. Confirm (a) is the greatest lower bound: Suppose there’s another lower bound (x) for ({a, a \vee b}). By definition, (x) must be ≤ (a) (since (a) is in the set). This means (a) is the largest possible lower bound.
4. Final conclusion: The infimum ((\wedge)) of ({a, a \vee b}) is exactly (a), so (a \wedge (a \vee b) = a).

Proof of (a \vee (a \wedge b) = a) #

This uses symmetric logic to the first proof:

1. Start with the infimum property: By definition, (a \wedge b) is the greatest lower bound of (a) and (b). This means it’s less than or equal to both elements, so (a \wedge b \leq a) holds.
2. Verify (a) is an upper bound: An upper bound of a set must be ≥ every element in the set. We know:
   • (a \geq a) (reflexivity of posets)
   • (a \geq a \wedge b) (from step 1)
     So (a) qualifies as an upper bound for the set ({a, a \wedge b}).
3. Confirm (a) is the least upper bound: Suppose there’s another upper bound (y) for ({a, a \wedge b}). By definition, (y) must be ≥ (a) (since (a) is in the set). This means (a) is the smallest possible upper bound.
4. Final conclusion: The supremum ((\vee)) of ({a, a \wedge b}) is exactly (a), so (a \vee (a \wedge b) = a).

内容的提问来源于stack exchange，提问作者gete