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This is a reference request.

Suppose $(G, \cdot)$ is a group and consider the structure on $G^{<\omega}$ where, for $\mathbf{p} = (p_1, \dots, p_n) \in G^{<\omega}$ and $\mathbf{q} = (q_1, \dots, q_m)$ we have

$\mathbf{p} \oplus \mathbf{q} = (p_1, p_2,..., p_n, q_1, q_2, ..., q_m)$

(i.e. concatenation) and

$\mathbf{p} \otimes \mathbf{q} = (p_1\cdot q_1,p_1\cdot q_2,\dots,p_1\cdot q_m,p_2\cdot q_1,p_2\cdot q_2,\dots,p_2\cdot q_m,\cdots ,p_n\cdot q_1,p_n\cdot q_2,\cdots,p_n\cdot q_m)$

What are structures of the from $(G, \oplus, \otimes)$ called? What is a good place to look to begin learning about them?

Thanks.

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That is not a ring: it does not have additive inverses. A semiring, maybe. in any case, why do you think this construction has already been studied? As a data point, I have never seen it before. –  Mariano Suárez-Alvarez Jul 21 '13 at 2:37
    
maybe you would like to see about the free monoid functor and monoidal categories? –  user40276 Jul 21 '13 at 2:48
    
Mariano: Thanks, I fixed the statement. As for why I think this has been studied, mainly because this seems like a very natural pair of operations and in my experience when you come across natural feeling operation on natural structures, someone has already thought about them. –  Nate Ackerman Jul 21 '13 at 3:20
    
User40276: While they are very interesting, I am specifically interested in how the above operations of $\oplus$ and $\otimes$ interact. –  Nate Ackerman Jul 21 '13 at 3:23
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