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so, you have, for any two members of the algebraic structure A and B and any nonnegative real values a, b:

two operations: * and +, such that

a*A + b*A = (a+b)*A is in the structure

A + B = B + A is in the structure

0*A + B = B

but there is no guarantee that X s.t.

X + A = B

is in the structure.

As an example, the set of 2-dimensional Cartesian vectors that are in the first quadrant (i.e., x>=0 and y>=0) has the properties that I want. You can add them, scale them, but if you try to subtract them, you might leave the first quadrant.

Thanks very much!

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I'm afraid this question isn't appropriate for MathOverflow. – Zev Chonoles Jan 26 '10 at 0:20
I don't think you can have fully specified what you want. Early in your question you make no mention of division or real numbers but later you say that monoids lack division by non-negative real numbers. – Dan Piponi Jan 26 '10 at 0:22
I tried to fix things-- maybe vector spaces are the closer analogy. – Neil Jan 26 '10 at 0:24
Why is this not appropriate for MathOverflow? – Neil Jan 26 '10 at 0:24
oh, and regarding division by real numbers, they are implied since if a*A forall nonnegative a is in the structure, so is (1/a)*A. – Neil Jan 26 '10 at 0:30
up vote 12 down vote accepted

If the structure in question is a subset of a vector space, like it is in your example, I would call it a convex cone.

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I believe it is called a module over a semiring. In your example, the semiring is commutative.

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Thank you! So, I would just call it a "module over a commutative semiring"? – Neil Jan 26 '10 at 0:38
Yes. Alternatively, you could say it is a module over the commutative semiring of nonnegative real numbers. – S. Carnahan Jan 26 '10 at 1:11

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