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Differential groupoids. (English) Zbl 0744.20055
General algebra, Proc. Conf., Vienna/Austria 1990, Contrib. Gen. Algebra 7, 283-290 (1991).
[For the entire collection see Zbl 0731.00007.]
The authors deal with the groupoids $$(G,.)$$ with the identities $$x.x=x$$, $$(x.y).(z.t)=(x.z).(y.t)$$, and $$x.(y.z)=x.y$$. Some connections between differential groupoids, differentials and differentiations are given. It is proved that the linearization $$Z\otimes \mathbb{D}$$ (= the tensor product of affine spaces $$Z$$ over the integers $$\mathbb{Z}$$ and the variety $$\mathbb{D}$$ of differential groupoids) is the variety of affine spaces over the dual integers $$\mathbb{Z}[d]$$ which is the abelian group $$\{r+sd: r,s\in\mathbb{Z}\}$$ with multiplication $$(r+sd)(t+ud)=rt+(st+ru)d$$. The notion of abstract differentiation and continuity of functions $$G\to G$$ in differential groupoids $$(G,.)$$ are introduced. Their properties justify their names.

##### MSC:
 20N02 Sets with a single binary operation (groupoids) 22A22 Topological groupoids (including differentiable and Lie groupoids) 08A30 Subalgebras, congruence relations 08A60 Unary algebras