On Small Doubling in Right-Ordered Groups and Baumslag-Solitar Groups-II

Recently, Mohan et al. [Results Math. 80 (2025), No. 4, 122] answered Freiman’s $3k-4$ conjecture in right-ordered groups under certain restrictions. In this paper, we take a step further by investigating the structure of nonempty subsets $S$ of a right-ordered group satisfying the small doubling condition $\mathrm{\mid }{S}^2\mathrm{\mid }=3\mathrm{\mid }S\mathrm{\mid }-3$. Moreover, we provide a complete characterization of all nonempty finite subsets $S$ of the Baumslag-Solitar group $\mathrm{B}\mathrm{S}(1,q)$ (with $q\in \mathbb{Z}$ and $q\mathrm{\ne }-1$) for which $\mathrm{\mid }{S}^2\mathrm{\mid }=3\mathrm{\mid }S\mathrm{\mid }-3$ and the identity element is the minimum of $S$.