Group structures of a function spaces with the set-open topology

In this paper, we find at the properties of the family $\lambda$ which imply that the space $C(X,\mathbb{R}^{\alpha})$ — the set of all continuous mappings on a Tychonoff space $X$ to the space $\mathbb{R}^{\alpha}$ with the $\lambda$-open topology is a semitopological group (paratopological group, topological group, topological vector space and other algebraic structures) under the usual operations of addition and multiplication (and multiplication by scalars). For example, if $X=[0,\omega_1)$ and $\lambda$ is a family of $C$-compact subsets of $X$, then $C_{\lambda}(X,\mathbb{R}^{\omega})$ is a semitopological group (locally convex topological vector space, topological algebra), but $C_{\lambda}(X,\mathbb{R}^{\omega_1})$ is not semitopological group.