3 ms·

Can you not make a "tropical Zn" with elements 0 - (M-1), and a "big" element, representing anything M or larger, including infinite values? I dont know that i

Can you not make a "tropical Zn" with elements 0 - (M-1), and a "big" element, representing anything M or larger, including infinite values? I dont know that it would be exactly analogous to normal Z[M+1], and might have useful properties for some kind of geometric or combinatorical modeling. (My hunch is any time you want to be capable of carrying a "crossed threshold" flag as well as a value, and have that cascade through calculation.) It would also model systems where you can get to infinity in finite steps, but can't traverse back. Not sure if those are useful in the abstract, though.

Someone 10y ago
IEEE 754 (https://en.wikipedia.org/wiki/IEEE_floating_point https://en.wikipedia.org/wiki/IEEE_floating_point) does that (for floating point, but the integer variant would be easily defined), with a few extensions. You need "small", too, for example, and "don't have the faintest idea" for when somebody subtracts "big" from "big". That's pragmatic, highly useful, but a nightmare for mathematicians. 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') Compiler writers may happily make matters worse by assuming those laws still hold, with the effect that the same computation may overflow or not on different CPUs, under different compilers, compiler settings, or even the same compiler in the same compilation run.
- llamaz 10y ago
  "almost groups" are called semigroups. Somehow people have written multiple books on the subjects.
  - Someone 10y ago
    Oops and thanks. I don't know how I managed to not fish that out of my memory while thinking whether half groups or incomplete groups existed. I even thought of the term quasigroup (or did I think of 'almost group'?) but unfortunately didn't google it, as it would have learnt me they exist.
- SomeStupidPoint 10y ago
  > 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

Someone 10y ago

IEEE 754 (https://en.wikipedia.org/wiki/IEEE_floating_point https://en.wikipedia.org/wiki/IEEE_floating_point) does that (for floating point, but the integer variant would be easily defined), with a few extensions. You need "small", too, for example, and "don't have the faintest idea" for when somebody subtracts "big" from "big". That's pragmatic, highly useful, but a nightmare for mathematicians. 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') Compiler writers may happily make matters worse by assuming those laws still hold, with the effect that the same computation may overflow or not on different CPUs, under different compilers, compiler settings, or even the same compiler in the same compilation run.

llamaz 10y ago
"almost groups" are called semigroups. Somehow people have written multiple books on the subjects.
- Someone 10y ago
  Oops and thanks. I don't know how I managed to not fish that out of my memory while thinking whether half groups or incomplete groups existed. I even thought of the term quasigroup (or did I think of 'almost group'?) but unfortunately didn't google it, as it would have learnt me they exist.
SomeStupidPoint 10y ago
> 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