Convolution $K$ -monoid

Suppose $𝑅$ is a $K$ -comonoid and $𝐵$ is a $K$ -monoid. Then the convolution monoid $𝑅 ⊸ 𝐵 : = K 𝖬 𝗈 𝖽 (𝑅, 𝐵)$ is in fact a $K$ -monoid, #m/thm/hopf i.e. the multiplication is $K$ -bilinear.

Proof

We prove linearity in the left argument, linearity in the right argument is similar. Let $𝑓, 𝑔, ℎ$ and $𝛼, 𝛽 \in K$ . Then for $𝑥 \in 𝑅$ , we have

[(𝛼 𝑓 + 𝛽 𝑔) * ℎ] (𝑥) = \sum (𝑥) (𝛼 𝑓 𝑥 (1) + 𝛽 𝑔 𝑥 (1)) \cdot ℎ 𝑥 (2) = \sum (𝑥) 𝛼 𝑓 𝑥 (1) \cdot ℎ 𝑥 (2) + \sum (𝑥) 𝛽 𝑔 𝑥 (1) \cdot ℎ 𝑥 (2) = [𝛼 (𝑓 * ℎ) + 𝛽 (𝑔 * ℎ)] (𝑥)

as required.

In the special case where $𝐵 = K$ , we get the dual $K$ -monoid $𝑅 *$ .

#state/tidy | #lang/en | #SemBr

Convolution K-monoid

Convolution $K$ -monoid