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.