Example

(1) Let S be a monoid with identity element u, so that the algebra $E = A^{(S)}$ of the monoid S over A admits the unit element $e_u$ (§ 2, no. 6); it has been seen on the other hand that E has canonically a coassociative cocommutative A-cogebra structure with a counit $\gamma$ such that $\gamma(e_s) = 1$ for all $s \in S$ (no. 1, Example 4 and no. 2, Examples 4, 5 and 6). The formula $c(e_s) = e_s \otimes e_s$ giving the coproduct shows also immediately that c is an algebra homomorphism. Thus a cocommutative bigebra structure has been defined on E and E, with this structure, is called the bigebra of the monoid S over A.

If T is another monoid with unit element v, $f : S \to T$ a homomorphism such that $f(u) = v$ and $f_{(A)} : A^{(S)} \to A^{(T)}$ the A-algebra homomorphism derived from $f$ (§ 2, no. 6), it is immediately verified that $f_{(A)}$ is a bigebra homomorphism.

(2) Let M be an A-module. The graded A-algebra (§ 6, no. 1) and graded A-cogebra (no. 1, Example 6) structures defined on S(M) define on this set a commutative cocommutative graded bigebra structure; for we have seen (no. 1, Example 6) that the coproduct on S(M) is an algebra homomorphism and it follows from the definition of the counit $\gamma$ (no. 2, Example 6) that $\gamma(1) = 1$ and that $\gamma$ is an algebra homomorphism of E into A.

(3) Let M be an A-module. We see as in Example 2 that the graded A-algebra (§ 7, no. 1) and graded A-cogebra (no. 1, Example 7) structures on $\bigwedge(M)$ define on this set an anticommutative anticocommutative skew graded bigebra structure.