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I have a set $A$ with two binary operations on it

$(A,*,\cdot)$

STRUCTURE A

  • $(A,*)$ is not commutative, is not associative, it has not an identity

  • $(A,\cdot)$ is a commutative group

  • $(a*b)\cdot c=(a\cdot c)*(b\cdot c)$

STRUCTURE B

  • $(A,*)$ is not associative, it has not an identity but it is commmutative

  • $(A,\cdot)$ is a commutative group

  • $(a*b)\cdot c=(a\cdot c)*(b\cdot c)$

Has one of these structures a name in the literature? Usually, even in the weakest kind of structures with two operations like semi-rings, we have the commutativity and the associativity of the additive operation but the structure A and B are really weaker than anything I've seen: these structure are like Near-Semi-fields with non-associative additive operation.

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  • $\begingroup$ If you're fine with artificially adding structure, Structure A can be equivalently described as a $\Bbb Z$-algebra (not necessarily associative) where the action of $\Bbb Z$ is multiplication in the underlying additive group. $\endgroup$ – user98602 Jul 1 '14 at 9:27
  • $\begingroup$ @MikeMiller where I can learn more about this? $\endgroup$ – MphLee Jul 1 '14 at 9:29
  • $\begingroup$ @MikeMiller By the way I don't get how this can help me. Isn't an R-algebra a special kind of R-module...but in my cases I have only two operations on the same set $\endgroup$ – MphLee Jul 1 '14 at 9:53
  • $\begingroup$ I'm not saying anything intelligent here. Any abelian group also has a unique $\Bbb Z$-module structure. So any of your structure A's also have a unique $\Bbb Z$-algebra structure. $\endgroup$ – user98602 Jul 1 '14 at 10:14
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    $\begingroup$ Pick a book that describes what a non associative algebra over a ring is. Profit. $\endgroup$ – Pedro Tamaroff Jul 1 '14 at 23:02

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