In two earlier articles, we provided sufficient conditions on (mass action) reaction network structure for the preclusion of multiple positive steady states in the context of what chemical engineers call the continuous flow stirred tank reactor. In such reactors, all species are deemed to be present in the effluent stream, a fact which played a strong role in the proofs. When certain species are deemed to be entrapped within the reactor, the questions that must be asked are more subtle, and the mathematics becomes substantially more difficult. Here we extend results of the earlier papers to semiopen reactors and show that very similar results obtain, provided that the network of chemical reactions satisfies certain weak structural conditions; weak reversibility is sufficient but not necessary.