Let $(S, \le)$ be a partially ordered finite set. Let $C$ the vector space with basis $\{e_{i,j} | i,j \in S, i \le j\}$, which turns out to be a coalgebra with comultiplication and counit given by: $$ \Delta (e_{i,j})= \sum_{k=i}^j e_{i,k} \otimes e_{k,j} \\ \epsilon(e_{i,j}) = \delta_{i,j}$$ I want to know:
- if $C$ is a simple coalgebra
- when $C$ is a simple coalgebra.
I know that a coalgebra is simple if, when I take $B \subseteq C$ a subcoalgebra, I obtain $B=0$ or $B=C$. So, I need to find what are the subcoalgebras of $C$; in other words, what are the subspaces $B$ of $C$ such that $\Delta(B) \subseteq B \otimes B$, but from there I am not sure how to continue, neither if it is the right path to solve questions 1 and 2.
Can anyone help me? Thanks in advance.