Name of an algebraic structure $(A,*,\cdot)$ weaker than semirings.

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.

• 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. – user98602 Jul 1 '14 at 9:27
• 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. – user98602 Jul 1 '14 at 10:14