In this article, we will introduce a concrete example (found in CWM) of interchanging distinct structures of algebra and order on a fixed set, where a set of monoid action infers a special type of order called lattice in a guise of T-algebra.
Proposition. A partially ordered set Q is called complete semi-lattice when every subset has supremum in Q. Let be a covariant power set functor on Set. For a set , maps to the set and maps each family of sets to the union set. a. is a monad on Set; b. It holds that each -algebra is a complete semi-lattice if the order is defined by and for each set , ; c. Conversely, every small complete semi-lattice is -algebra;
Proof of a. For the unit , we see that it commutes the diagram:
while for the product , we see that it commutes the following diagram:
Note that every composition of the maps used in the left diagrams are horizontal compositions, where the functor is regarded as natural identity . On the right side of the diagram, we depicted an instance of element mapping considering as an element of powerset and as such.
Proof of b. First we show that is partially ordered.
The antisymmetry trivially holds. The reflexivity is by definition of -algebra . The transitivity is also followed by definition. Assume and , then it is shown that as in the following commutative diagram:
which concludes that is partially ordered.
To see is a complete semi-lattice, we prove indeed defines for each subset . For any , we have and if there exists such that , then by definition; therefore .
Proof of c. Let be a complete semi-lattice. Then commutes the diagram:
while holds. Hence is a -algebra with the structure map .