Let be an algebraic semilattice. (That is, an commutative semigroup in which every element is idempotent.) Recall the natural partial order on given by if and only if . Show that any two elements and of have a greatest lower bound with respect to , which is .
Conversely, suppose is an order semilattice. (That is, we have a partial order on such that any two elements have a greatest lower bound.) Show that the operator on makes into an algebraic semilattice, whose natural order coincides with .
Suppose is an algebraic semilattice, and let . Now and , so that and . Now suppose and . Then and , so that , and we have . So is a greatest lower bound of and with respect to the natural order.
Now suppose is an order semilattice, and denote the greatest lower bound of and by . We claim that is associative. To that end, let . Note that , , and . So , and we have . Similarly, , so that , and we have . Since is antisymmetric, . So is a semigroup under . Certainly this operator is commutative and associative, so that is an algebraic semilattice. Now let denote the natural order on . If , we have , so that . Conversely, if , then , and thus .