A bounded semilattice is an idempotent commutative monoid. We can define equivalence classes of idempotent matrices and form an abelian monoid formula 51. The merge function is commutative, associative, and idempotent. This makes "S" an idempotent, commutative quasigroup. This relation gives a bijection between involutory matrices and idempotent matrices. It follows that every nonempty periodic semigroup has at least one idempotent. Finally, an inverse semigroup with only one idempotent is a group. There are two nontrivial idempotent elements given by and . In particular, the Karoubi envelope of a category splits every idempotent morphism. Then, these can be identified with idempotent matrices in some ring of matrices formula 50. Algebraic closings are operators in "L" that are idempotent, increasing, and extensive. Algebraic openings are operators in "L" that are idempotent, increasing, and anti-extensive. The identity matrix is the only idempotent matrix with non-zero determinant. One example of an idempotent element is a projection in linear algebra. Such an element "p" is called a "separability idempotent", since it satisfies formula 7.