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I need 2 variable functions that are closed under addition, multiplication, and scalar multiplication. for example; for 2 variables ($a\text{ and }b$),
$$ f(a, b) = e^{a^2}+e^{b^2}+e^{a^2+b^2} $$ thanks.

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A function by itself cannot be closed under an operation. Do you mean that you are looking for a set of functions of two variables that is closed under addition, multiplication and scalar multiplication? –  joriki Dec 15 '12 at 7:33
yes,that's right,maybe set of trigonometric functions or something like that. –  mohamad Dec 15 '12 at 8:44
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2 Answers

Let $A,B$ be two sets and $R$ a ring. Then the set of all functions $A\times B\to R$ is an $R$-module, i.e. closed under addtion, multiplication and scalar multiplication with elements of $R$.

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The set $\{f\}$ with $f(x,y)=0$ is closed under addition, multiplication and scalar multiplication.

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consider our function is f(a,b)=exp(a^2)+exp(b^2),it is obvious that this set of function isn't close under multiplication. –  mohamad Dec 15 '12 at 10:30
@mohamad: I don't understand what you mean by "our function is $f(a,b)=\exp(a^2)+\exp(b^2)$". Also, this information was apparently missing in the question, which contained a different function, which was merely provided as an example. If certain functions are assumed to be included in the set, please clarify the question accordingly. –  joriki Dec 15 '12 at 10:36
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