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The ring-like structures that can be defined on the ground supermanifolds R 1|1 and C 1|1 are classified up to equivalence in the category of smooth and complex Berezin-Kostant-Leites-Manin supermanifolds. It is proved that there are three different such equivalence classes in the real case, whereas there are two for the complex field. The corresponding module structures-defined componentwise on the product of k copies of R 1|1 or C 1|1 -are also classified up to equivalence. The notions of linearity and bilinearity are reviewed and used to define Heisenberg-like super group structures. It turns out that there are three non-isomorphic real such super groups, whereas only two over the complex field. The use of the appropriate exponential maps introduces the possibility of defining Heisenberg-like super group structures on the product of k copies of the ground supermanifold, with an appropriate super circle. The corresponding classification is also obtained.

It is shown how, the classification of those Lie superalgebras having u2 as its underlying Lie algebra, and further restricted by the fact that the representation of u2 into its odd module is the adjoint representation, leads to at least ten different natural superhomogeneous models for Minkowski superspacetime.

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It is proved that up to isomorphism there is only one (2, 2)-dimensional supertorus associated to a nontrivial representation of its underlying 2-torus, and that it has nontrivial odd brackets. This supertorus is obtained by finding out first a canonical form for its Lie superalgebra, and then using Lie's technique to represent it faithfully as supervector fields on a supermanifold. Those supervector fields can be integrated, and through their various integral flows the composition law for the supergroup is straightforwardly deduced. It turns out that this supertorus is precisely the supergroup described by Guhr (1993) following a formal analogy with the classical unitary group U(2) but with no further intrinsic characterization.2000 Mathematics Subject Classification: 17B70, 58A50, 58C50, 81R05.