$(A note on the $\lambda$-super socle in $C(X

Let $S_\lambda(X) = \{f \in C(X) : |Coz(f)| \lambda\}$, where $\lambda $ is a regular cardinal number with $\lambda\geq |X|$. In this paper $S_\lambda(X)$ is characterized both algebrically and topologically when $X$ is Maximal $\lambda$-compact. We show that the intersection of ideals $O_x$ where $x$ runs through non $\lambda$-isolated points in $X$ is the $\lambda$-super socle of $C(X)$ if and only if every open set containing all non $\lambda$-isolated points is co-$\lambda$ set (an open set $G$ is co-$\lambda$ set if $|X \ G| \lambda)$. Finally, it is proved that $\lambda$-Fréchet z-filter is the intersection of $\lambda$-essential z-filters.