[TYPES] What algebra am I thinking of?

[Tadeusz Litak]

Hi Neel,

see § 4.3 and especially Exercise 4.13 in Haghverdi and Scott's "Geometry of Interaction and the Dynamics of
Proof Reduction: a tutorial".

Best regards,
t.


On 28/03/18 12:21, Neel Krishnaswami wrote:
> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ]
>
> Hi Phil,
>
> I learned about this from Martin Escardo, and he told me the observation
> about monoidal closure is in Lawvere's paper "Metric Spaces, Generalized
> Logic, and Closed Categories".
>
> One further observation I have not seen written down anywhere is that
> the natural numbers also support a trace (ie, feedback) operator, since
> it is also the case that x + y ≥ z + y if and only if x ≥ z.
>
> This means that the Geometry of Interaction construction can be applied
> to the natural numbers, which yields a preorder of pairs of natural
> numbers (n,m). In this case, two objects have a map between them if
> one object is greater than the other, viewing each object (n, m) as the
> signed integer n - m.
>
> As a result, I've sometimes wondered if the Int construction got its
> name as a bit of a joke!
>
> Best,
> Neel
>
>
> On 27/03/18 21:02, Philip Wadler wrote:
>> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ]
>>
>>
>>
>> Thanks to all for further replies, and especially to Neel for his
>> observation about monus and monoidal closure. Neel, is this written down
>> anywhere? Cheers, -- P
>>
>> .   \ Philip Wadler, Professor of Theoretical Computer Science,
>> .   /\ School of Informatics, University of Edinburgh
>> .  /  \ and Senior Research Fellow, IOHK
>> . http://homepages.inf.ed.ac.uk/wadler/
>>
>> On 27 March 2018 at 13:39, Neel Krishnaswami <nk480 at cl.cam.ac.uk> wrote:
>>
>>> On 27/03/18 15:27, Philip Wadler wrote:
>>>
>>> Thank you for all the replies, and for reminding me of bags,
>>>> rigs/semirings, and modules.
>>>>
>>>> In a rig, what corresponds to the notion of monus?
>>>>     _∸_ : ℕ → ℕ → ℕ
>>>>     m          ∸ zero      =  m
>>>>     zero      ∸ (suc n)  =  zero
>>>>     (suc m) ∸ (suc n)  =  m ∸ n
>>>> Is monus defined for every rig? (While zero and suc are defined in every
>>>> rig, it's not immediately obvious to me that definition by pattern
>>>> matching
>>>> as above makes sense in an arbitrary rig.) What are the algebraic
>>>> properties of monus?
>>>>
>>>
>>> The natural numbers Nat, ordered by , form a posetal symmetric
>>> monoidal closed category. Here, the objects are the natural
>>> numbers, and Hom(n, m) = { ∗ | n ≥ m }
>>>
>>> The tensor product is given by:
>>>
>>>    x ⊗ y ≜ x + y
>>>
>>> The monoidal closed structure arises from saturating subtraction:
>>>
>>>    y ⊸ z ≜ z - y
>>>
>>> It's easy to show that Hom(x ⊗ y, z) = Hom(x, y ⊸ z) since
>>> x + y ≥ z if and only if x ≥ z - y.
>>>
>>> (This is all given in Lawvere's paper "Metric Spaces, Generalized
>>> Logic, and Closed Categories".)
>>>
>>> So it seems like the natural thing to do would be to say that a rig
>>> has a monus when it is monoidal closed with respect to its addition,
>>> defining x ≥ z to hold when there exists a y such that x = y + z.
>>>
>>> Best,
>>> Neel
>>>
>>>
>>>
>>> -- 
>>> Neel Krishnaswami
>>> nk480 at cl.cam.ac.uk
>>>
>>>
>>>
>>>
>>> The University of Edinburgh is a charitable body, registered in
>>> Scotland, with registration number SC005336.