Simple pp expansions of a quasivariety K are precisely the quasivarieties M such that the forgetful functor U from M to K is well-defined and M is isomorphic to a mono-reflective subcategory of K.
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A categorical description of simple Beth companions
Simple pp expansions of a quasivariety K are precisely the quasivarieties M such that the forgetful functor U from M to K is well-defined and M is isomorphic to a mono-reflective subcategory of K.