Some variants of $\omega$-balancedness in semitopological groups

We introduce the notion of $\theta$-$\omega$-balanced and $\delta\theta$-$\omega$-balanced semitopological groups and prove that both classes, the class of $\theta$-$\omega$-balanced semitopological groups and the class of $\delta\theta$-$\omega$-balanced semitopological groups, are closed under taking subgroups and products. We show that a regular (Hausdorff, $T_{1}$) semitopological group $G$ admits a homeomorphic embedding as a subgroup into a product of regular (Hausdorff, $T_{1}$) semitopological groups with a $\theta$-base if and only if $G$ is $\theta$-$\omega$-balanced and $Ir(G)\leq\omega$ $(Hs(G)\leq\omega,~Sm(G)\leq\omega)$. Similarly, a regular (Hausdorff, $T_{1}$) semitopological group $G$ admits a homeomorphic embedding as a subgroup into a product of regular (Hausdorff, $T_{1}$) semitopological groups with a $\delta\theta$-base if and only if $G$ is $\delta\theta$-$\omega$-balanced and $ Ir(G)\leq\omega$ ($ Hs(G)\leq\omega$, $Sm(G)\leq\omega $). Moreover, we give a characterization of a semitopological group that can be embedded as a subgroup into a product of regular (Hausdorff, $T_{1}$) quasi-developable semitopological groups. At the end of this article, we give the relationship between weakly $\omega$-balancedness, $M$-$\omega$-balancedness and $\omega$s-balancedness, and their examples.