## Equivalence of algebraic and order semilattices

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 .

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