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@BenjaminSteinberg If Im right, an ISP class is equivalent to a quasivariety (class closed under isomorphism, substructures and products), HSP class is equivalent to a variety (class closed under homomorphic images, substructures and products). And $Fr_\omega(V)$ is the free structure of the class with a basis of cardinal $\omega$.
@BenjaminSteinberg can you help me with an example of ISP class (V) that is not determined by Frω(V). Also if you can help me with a hint to prove that an HSP class is determined by Frω(V) will be very usefull.
@EmilJeřábek can you help me with an example of ISP class ($\mathbf{V}$) that is not determined by $Fr_\omega(\mathbf{V})$. Also if you can help me with a hint to prove that an HSP class is determined by $Fr_\omega(\mathbf{V})$ will be very usefull.
