Partial Order Approach
Let A be a set with a partial order ≤, and let x and y be two elements in A. An element z of A is the meet (or greatest lower bound or infimum) of x and y, if the following two conditions are satisfied:
- z ≤ x and z ≤ y (i.e., z is a lower bound of x and y).
- For any w in A, such that w ≤ x and w ≤ y, we have w ≤ z (i.e., z is greater than or equal to any other lower bound of x and y).
If there is a meet of x and y, then indeed it is unique, since if both z and z′ are greatest lower bounds of x and y, then z ≤ z′ and z′ ≤ z, whence indeed z = z′. If the meet does exist, it is denoted x ∧ y. Some pairs of elements in A may lack a meet, either since they have no lower bound at all, or since none of their lower bounds is greater than all the others. If all pairs of elements have meets, then indeed the meet is a binary operation on A, and it is easy to see that this operation fulfils the following three conditions: For any elements x, y, and z in A,
- a. x ∧ y = y ∧ x (commutativity),
- b. x ∧ (y ∧ z) = (x ∧ y) ∧ z (associativity), and
- c. x ∧ x = x (idempotency).
Read more about this topic: Join And Meet
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