The operation ⊕ is associative and commutative, and
The only element which could potentially act as an identity element is 0, since an identity e must satisfy e⊕e = e. This yields the equation , but if e is nonzero that implies , so e could only be zero. Unfortunately 0 does not work as an identity element after all, since 0⊕(−1) = 1. This does indicate, however, that if the operation ⊕ is restricted to nonnegative real numbers, then 0 does act as an identity. Consequently, the operation ⊕ acting on the nonnegative real numbers forms a commutative monoid.
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