Given some non-empty set $X$, does the set of positive definite kernels on $K\_X$ form a ring with pointwise addition and multiplication. I am convinced it does not as surely if $k \in K\_X$ then we would require that $-k \in K\_X$ also, which I don't think is true? 


I ask because Hagan (the user) seems to think it does in this question: [The Polynomial Kernel](https://mathoverflow.net/questions/31030/the-polynomial-kernel)


If not, does it even form an abelian group with respect to any other operation? Or better yet, a ring? 



Does the set of positive definite kernels on some set X form a ring?
To make it answered (cw): 



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> It's not stable under taking $x\mapsto -x$ so is not a ring, as it's not a group under addition.
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> 
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