Can you choose -1 as the multiplicative unit? And what is a positive number? If one starts with the cyclic group of integers and want to introduce multiplikation the ordinare choice of multiplicative identity is the generator 1. But since 1 and its inverse -1 is sort of the same in the cyclic group, there is an automorphism that exchange these it should be possible to use -1 as multiplicative unit? 
How can "a positive number" be defined? In the ordinary setting starting with the integers as a cyclic (additive) group it seems that once you choose the multiplicative unit there is a split in positive and negative numbers.
In what sort of structures does it make sense? For all Z/nZ it seems possible to do with one sort of representative, so for practical purposes we could do arithmetic on the whole number in Z/nZ with a sufficiently large n. 

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*There is an automorphism of $\mathbb{Z}$ the abelian group that exchanges $1$ and $-1$, but it doesn't respect multiplication: that is, it isn't an automorphism of $\mathbb{Z}$ the ring, which has no nontrivial automorphisms.

*Positivity comes from an order, which in general neither exists nor is unique. $\mathbb{Z}/n\mathbb{Z}$ does not have an order (compatible with its group structure). 
