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.

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*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 .

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*Proof of *c. Let be a complete semi-lattice. Then commutes the diagram:

while holds. Hence is a -algebra with the structure map .

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