Join and Meet - Universal Algebra Approach

Universal Algebra Approach

By definition, a binary operation ∧ on a set A is a meet, if it satisfies the three conditions a, b, and c. The pair (A,∧) then is a meet-semilattice. Moreover, we then may define a binary relation ≤ on A, by stating that x ≤ y if and only if x ∧ y = x. In fact, this relation is a partial order on A. Indeed, for any elements x, y, and z in A,

• x ≤ x, since x ∧ x = x by c;
• if x ≤ y and y ≤ x, then x = x ∧ y = y ∧ x = y by a; and
• if x ≤ y and y ≤ z, then x ≤ z, since then x ∧ z = (x ∧ y) ∧ z = x ∧ (y ∧ z) = x ∧ y = x by b.

Note that both meets and joins equally satisfy this definition: a couple of associated meet and join operations yield partial orders which are the reverse of each other. When choosing one of these orders as the main ones, one also fixes which operation is considered a meet (the one giving the same order) and which is considered a join (the other one).