I recently learned about the tensor product of Abelian groups, which can be used to define the concept 'ring.' In particular, a ring is just a monoid object in the monoidal category of Abelian groups, with tensor product for the monoidal product.

I'm wondering whether its possible to describe near-rings in a similar fashion. Here are my thoughts. Given groups $G$ and $I$, a set $H$, and a function $\varphi : G \times H \rightarrow I$, let us say that $\varphi$ is a levomorphism iff for every $h \in H$, the function $g \in G \mapsto \varphi(g,h)$ is a homomorphism. So $\varphi$ is required to be a homomorphism in the left argument, for any $h \in H$ that is put into the right argument.

Now given a group $G$ and a set $H$, by a tensor semiproduct of $G$ and $H$, let us mean a group $G \lhd H$ together with a levomorphism $$\varphi : G \times H \rightarrow G \lhd H$$

satisfying the obvious universal property. Namely that, that for any group $I$ and any levomorphism $\psi : G \times H \rightarrow I$, there exists a unique homomorphism $\psi' : G\lhd H \rightarrow I$ such that $\psi = \psi' \circ \varphi$.

Question. Do tensor semiproducts of groups actually exist, and if so, can we define near-rings as some kind of a monoid object in the category of groups?

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    $\begingroup$ On tensor semiproducts: did you try the obvious generators-and-relations construction? $\endgroup$ – Zhen Lin Sep 9 '14 at 15:56

Yes, these tensor semiproducts exist, just let $$G \lhd H := \coprod_{h \in H} G.$$ Near-rings can be desribed via associative maps $G \lhd G \to G$. I have also explained this here.


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