Remark

a) It is easy to show that for every $K$-algebra homomorphism $\varphi$ of $\Theta(X)$ into $K$, there exists a unique element $x$ of $X$ such that $\varphi(f) = f(x)$ for all $f \in \Theta(X)$.

2) Let $X$ and $Y$ be two finite $\Gamma$-sets. Let $\mathfrak{T}_\Gamma(X, Y)$ be the set of mappings $u$ of $X$ into $Y$ such that $u(\sigma x) = \sigma u(x)$ for all $\sigma \in \Gamma$ and all $x \in X$. For $u \in \mathfrak{T}_\Gamma(X, Y)$ we define a $K$-algebra homomorphism $u^*: \Theta(Y) \to \Theta(X)$ by $u^*(f) = f \circ u$. For every homomorphism $\Psi$ of $\Theta(Y)$ into $\Theta(X)$ there exists a unique element $u$ of $\mathfrak{T}_\Gamma(X, Y)$ such that $\Psi = u^*$.