The BOB-property of locally solid Riesz spaces was introduced by Labuda in 1985. In this study, we investigate certain aspects of this property. For example, it is stated that the Dedekind completion of a locally solid Riesz space E has BOB-property iff E has the same property. It is also observed that under some sufficient conditions, the property passes to the topological completion of the underlying space. We give a characterization of BOB-property of the space \(L_{b}\left( L,F\right) \) , consisting of all order bounded operators from a Riesz space L into a Dedekind complete locally convex-solid Riesz space F, in terms of that of the range space F. Moreover, we introduce topologically b-order bounded operators by means of topologically b-order bounded sets and we investigate the properties of these operators. Finally, we investigate the order structure of the space of generalized b-weakly compact operators introduced by Altin-Machrafi (2022). It is for example stated under natural conditions on the domain and range spaces that whenever the range space enjoys BOB-property then the space of order bounded generalized b-weakly compact operators is stable under the modulus operation.
