Introduction to Mathematical Logic, Sixth Edition
《Introduction to Mathematical Logic, Sixth Edition》 · 第 192 页 · Chapter 4，Section 4.1.8「Exercises」，Exercise 3
Let $A$ be a bialgebra, in the sense of the previous exercises with comultiplication map $\Delta$ and counit $\varepsilon$. Let $S \colon A \to A$ be an antiautomorphism ($S(xy) = S(y)S(x)$ for all $x, y \in A$). Then $S$ is called an antipode if $\mu((S \otimes I_A)(\Delta(a))) = \varepsilon(a)1$ and $\mu((I_A \otimes S)(\Delta(a))) = \varepsilon(a)1$ for all $a \in A$, where $\mu \colon A \otimes A \to A$ is the multiplication map. A bialgebra with an antipode is called a Hopf algebra.