If I wanted to reduce hysteresis losses in a system, I know that I would need to choose a ferromagnetic core made from soft magnetic material. Do soft magnetic materials have smaller permeability compared to hard magnetic materials? And is the permeability directly related to the area of the hysteresis loop?
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A magnetically hard material has a large coercive field, so a wide hysteresis loop. It is what hard and soft means in this context.
Permeability is not well defined when there is hysteresis, as the magnetization is not proportional to the external field. When hysteresis can be neglected, such soft magnets have huge values of the permeability (for example mu-metal).