I like the taxonomy of animal-human relationships as a model for asking how humans could relate to AI in the future. It's useful for framing the problem. However, I don't think that any existing relationship model would hold true for a superintelligence. We keep lapdogs because we have emotional reactions to animals, and to some extent because we need to take care of things. Would an AI? We tolerate dust mites in our eyelashes because we don't notice them, and can't do much about them anyway. Is that true for an AI? What does such an entity want or need, what are their motivations, what really pisses them off? Or, do any of those concepts hold meaning to them? The relationship between humans and a superintelligent AGi just can't be imagined.