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We establish a relation between Lipschitz operator ideals and linear operatorideals, which fits in the framework of Galois connection between lattices. Weuse this relationship to give a criterion which allow us to recognize when aBanach Lipschitz operator ideal is of composition type or not. Also, weintroduce the concept of minimal Banach Lipschitz operator ideal, which haveanalogous properties to minimal Banach operator ideals. Also we characterizeminimal Banach Lipschitz operator ideals which are of composition type andpresent examples which are not of this class.

Galois connection between Lipschitz and linear operator ideals and minimal Lipschitz operator ideals