##  [Bialgebra](/bialgebra) 

  ##  [Bialgebra](https://algebra.quantumdictionary.io/bialgebra-0) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A bialgebra is a vector space or module equipped simultaneously with an associative unital algebra structure (multiplication and unit) and a coassociative counital coalgebra structure (comultiplication and counit), together with compatibility conditions that make the comultiplication and counit algebra homomorphisms (equivalently, multiplication and unit are coalgebra homomorphisms).