Join and Meet - Equivalence of Approaches

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.