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
Famous quotes containing the words partial, order and/or approach:
“You must not be partial in judging: hear out the small and the great alike; you shall not be intimidated by anyone, for the judgment is God’s.”
—Bible: Hebrew, Deuteronomy 1:17.
“This is one of the worst speeches I’ve ever seen. No one will listen except the Mobil P.R. man. List what we want to say M arrange items in order of priority M then say them plainly and bluntly.... Hit hard and early. Don’t apologize or evade tough issues.”
—Jimmy Carter (James Earl Carter, Jr.)
“The minute you try to talk business with him he takes the attitude that he is a gentleman and a scholar, and the moment you try to approach him on the level of his moral integrity he starts to talk business.”
—Raymond Chandler (1888–1959)