Finite coalgebra pieces of a multiplicative coordinate algebra

For $a\in A$, write $\mathrm{\Delta }(a)={\sum }_{i=1}^s{v}_i\otimes w_i$ with the $w_i$ linearly independent. Coassociativity, followed by coefficient functionals on the third tensor factor, shows $\mathrm{\Delta }(v_i)\in V\otimes A$, where $V$ is the span of the $v_i$. The counit gives $a\in V$. Adding the finitely many resulting spaces gives a finite-dimensional right coideal $V$ containing the prescribed subset. In a basis $v_1,\dots ,v_d$ write $\mathrm{\Delta }(v_j)={\sum }_i{v}_i\otimes c_{ij}$. Coassociativity and the counit give $\mathrm{\Delta }(c_{ij})={\sum }_{\mathrm{\ell }}{c}_{i\mathrm{\ell }}\otimes c_{\mathrm{\ell }j}$ and $ϵ(c_{ij})={\delta }_{ij}$. Their finite span $C$ is a subcoalgebra; applying $ϵ$ on the first factor gives $v_j={\sum }_iϵ(v_i)c_{ij}$, so $V\subset C$.

Inside $K[M]$, any finite-dimensional subcoalgebra $C_K$ is spanned by monomials. Indeed, if $\sum a_m{e}_m\in C_K$, apply the $e_m$ coefficient functional to the second tensor factor of its comultiplication; this produces $a_m{e}_m\in C_K$. All monomials appearing in a finite basis therefore belong to $C_K$ and span it. The dual basis consists of orthogonal idempotents, with sum the unit, because $\mathrm{\Delta }(e_m)=e_m\otimes e_m$ and $ϵ(e_m)=1$. Hence $(C_K{)}^{\ast }\cong K^d$ where $d=\dim C$, and finite dimension identifies this dual with $C^{\ast }{\otimes }_{k}K$.