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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.

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