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$S$ 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 ?

EDIT

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.

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    @Rajesh D: Knowing about them may or may not give it the structure that you would like. But *for certain*, **not** knowing about them will **not** help you. If you think it's better to waste your time trying to reinvent the wheel (and perhaps failing, perhaps succeeding), or worse, trying to construct a perpetual motion machine, then all power to you, stop asking whether the structure has a formalism or not, and just go bang your head against the wall on your own. The advantage of the formalism is simply that *others* have already broken through the wall, so that might save you some headaches.2010-12-02

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I think what you are describing is a semirings without identity: a semiring would be aset $R$ with two operations, $+$ and $\cdot$, such that $(R,+)$ is a commutative monoid, $(R,\cdot)$ is a commutative monoid, $\cdot$ distributes over $+$, and the neutral element relative to $+$ is a zero element relative to $\cdot$. The only difference between this and your structure is the existence of an identity for the operation *. Some authors allow semirings to not have a multiplicative identity, which would be exactly what you have.