> You lose invariants such as x+y-y=x (associativity and communicativity, in general), so you're no longer talking of a group (I don't know of research on 'almo

> You lose invariants such as x+y-y=x (associativity and communicativity, in general), so you're no longer talking of a group (I don't know of research on 'almost groups') Well, you actually just lose general inverses, so there's no sensible '-' operation that can just be turned in to '+ (-x)', where -x is the inverse of x under +. It's similar to the case where you have x * y / y = x except in the case where y = 0 (because there's no 0^-1). (As someone else pointed out, this puts you in to a semi-ring rather than a ring, which is what the general tropical semi-ring is. [1]) You can still have elements for which x + y - y = x is true, and usually that's a well-defined sub-space of x,y combinations. Proofs then usually use a case analysis: either we're in the subspace where x + y - y = x holds, or else we can use a property of not being in that space to derive a different but still useful conclusion. In some senses, it acts as a Maybe type, and can perform arithmetic without having to unpack that Maybe-ness. [1] https://ncatlab.org/nlab/show/tropical+semiring https://ncatlab.org/nlab/show/tropical+semiring