$S$ is a collection of disjoint sets. $(S,\cdot)$ is a commutative monoid and $(S,*)$ is a commutative semigroup. The identity element $e$ of $(S,\cdot)$ is the zero element of $(S,*)$. The monoid $(S,\cdot)$ does not have a zero element. The binary operator $'*'$ is distributive over the binary operator $'\cdot'$. What can be said about such an algebraic structure ? What could be its usefulness ?
There is a third binary operator $\otimes$ with which $(S,\otimes)$ is a commutative semigroup and the operator $'\otimes'$ is distributive over $'\cdot'$. $(S,\otimes)$ does not have a zero (absorption) element.
Motivation behind the question :
I had something and wanted to check where it fits in a formalism and it happened to turn out like this. I have a naive question, why should i even bother about such a formalism like semigroup,semiring etc., how is it useful.
PS: I thought it is OK/appropriate to edit a question to add to ask more on the same subject. Please let me know if it is not OK.I can make it as another question.