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April 10th, 2014, 08:45 PM   #1
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Name for a kind of set

Hi,

I came across a problem that is solved by using set with the following properties

Set S is an abelian semigroup under an operation (say +)
Set S is an semigroup under an operation (say .)
operation . is distributive over +. i.e. if a, b, c are members of set S, then
a.(b+c) = (a.b) + (a.c)

Is there a name for this kind of a set?
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April 13th, 2014, 06:56 PM   #2
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If it had a 0, and a 1, it would be a semi-ring. Some authors also use this term for semi-rings that do not possess a multiplicative identity, and some do not even require a 0:
compare:

Semiring - Wikipedia, the free encyclopedia
Semiring -- from Wolfram MathWorld

If we require the distributivity just be one-sided, but still have a 0, which in addition is an "absorbing element" (0.a = 0, or a.0 = 0, depending on "which sidedness we have") we have a near-semiring.

Many of these structure arise as certain kinds of functions, or by considering structures based on the natural numbers.
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