Equivalence of Approaches
If (A,≤) is a partially ordered set, such that each pair of elements in A has a meet, then indeed x ∧ y = x if and only if x ≤ y, since in the latter case indeed x is a lower bound of x and y, and since clearly x is the greatest lower bound if and only if it is a lower bound. Thus, the partial order defined by the meet in the universal algebra approach coincides with the original partial order.
Conversely, if (A,∧) is a meet-semilattice, and the partial order ≤ is defined as in the universal algebra approach, and z = x ∧ y for some elements x and y in A, then z is the greatest lower bound of x and y with respect to ≤, since
- z ∧ x = x ∧ z = x ∧ (x ∧ y) = (x ∧ x) ∧ y = x ∧ y = z
and therefore z ≤ x. Similarly, z ≤ y, and if w is another lower bound of x and y, then w ∧ x = w ∧ y = w, whence
- w ∧ z = w ∧ (x ∧ y) = (w ∧ x) ∧ y = w ∧ y = w.
Thus, there is a meet defined by the partial order defined by the original meet, and the two meets coincide.
In other words, the two approaches yield essentially equivalent concepts, a set equipped with both a binary relation and a binary operation, such that each one of these structures determines the other, and fulfil the conditions for partial orders or meets, respectively.
Read more about this topic: Join And Meet
Famous quotes containing the word approaches:
“These were not men, they were battlefields. And over them, like the sky, arched their sense of harmony, their sense of beauty and rest against which their misery and their struggles were an offence, to which their misery and their struggles were the only approaches they could make, of which their misery and their struggles were an integral part.”
—Rebecca West (1892–1983)