Definition 3

Let C be a cogebra over the field K and $c$ its coproduct. A subcogebra of C is any linear subspace $C_1$ of C such that $c(C_1)$ is contained in the canonical image of $C_1\otimes_KC_1$ in $C\otimes_KC$.

Let $j$ be the canonical injection of $C_1$ into C. By this definition, there exists a unique linear mapping $c_1: C_1\rightarrow C_1\otimes_KC_1$ such that we have

this relation means that $j$ is a morphism of cogebras from $(C_1, c_1)$ to $(C, c)$ (III, §11, No. 1, p. 574).

If C is coassociative, then $C_1$ is coassociative. If C is cocommutative, then so is $C_1$. If C has a counit $\varepsilon$, then the restriction of $\varepsilon$ to $C_1$ is a counit of $C_1$.