monoid bialgebra

Let $G$ be a monoid and $k$ a field. Consider the vector space $kG$ over $k$ with basis $G$. More precisely,

$$kG=\{f:G\to k|f(g)=0\text{ for almost all }g\in G\}.$$

We identify $g\in G$ with a function $f_g:G\to k$ such that $f_g(g)=1$ and $f_g(h)=0$ for $h\ne g$. Thus, every element in $kG$ is of the form

$$\sum _{g\in G}{\lambda }_{g}g,$$

for ${\lambda }_g\in k$. The vector space $kG$ can be turned into a $k$-algebra, if we define multiplication as follows:

$$g\cdot h=gh,$$

where on the right side we have a multiplication in the monoid $G$. This definition extends linearly to entire $kG$ and defines an algebra structure on $kG$, where neutral element of $G$ is the identity in $kG$.

Furthermore, we can turn $kG$ into a coalgebra as follows: comultiplication $\mathrm{\Delta }:kG\to kG\otimes kG$ is defined by $\mathrm{\Delta }(g)=g\otimes g$ and counit $\epsilon :kG\to k$ is defined by $\epsilon (g)=1$. One can easily check that this defines coalgebra structure on $kG$.

The vector space $kG$ is a bialgebra with with these algebra and coalgebra structures and it is called a monoid bialgebra.