### A characterization of ...-classes of semigroups as partial groupoids.

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We first recall Malgrange’s definition of $D$-groupoid and we define a Galois $D$-groupoid for $q$-difference equations. Then, we compute explicitly the Galois $D$-groupoid of a constant linear $q$-difference system, and show that it corresponds to the $q$-difference Galois group. Finally, we establish a conjugation between the Galois $D$-groupoids of two equivalent constant linear $q$-difference systems, and define a local Galois $D$-groupoid for Fuchsian linear $q$-difference systems by giving its realizations.

We discuss a concept of loopoid as a non-associative generalization of Brandt groupoid. We introduce and study also an interesting class of more general objects which we call semiloopoids. A differential version of loopoids is intended as a framework for Lagrangian discrete mechanics.