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Sat, 25 Jan 2025 03:26:56 -0800 (PST) MIME-Version: 1.0 From: Bertalan Pecsi Date: Sat, 25 Jan 2025 12:26:45 +0100 X-Gm-Features: AWEUYZl222Js4fTfRJKy45x7nzWp1v7GVBoWgNutdD__kBvXEoZxxZrxxavgo48 Message-ID: Subject: On why addition of ultrafilters over N is not commutative To: Categories mailing list X-Mimecast-Spam-Score: 0 X-Mimecast-MFC-PROC-ID: 8qvXGZFBMO6EBCL0x4rWcP1YOyqD6FhCBjqPp8ExJvU_1737804418 X-Mimecast-Impersonation-Protect: Policy=MQ - Tag Header Only on Default Settings;Similar Internal Domain=false;Similar Monitored External Domain=false;Custom External Domain=false;Mimecast External Domain=false;Newly Observed Domain=false;Internal User Name=false;Custom Display Name List=false;Reply-to Address Mismatch=false;Targeted Threat Dictionary=false;Mimecast Threat Dictionary=false;Custom Threat Dictionary=false Content-Type: multipart/alternative; boundary="000000000000f18555062c861e0a" X-EOPAttributedMessage: 0 X-EOPTenantAttributedMessage: 82c514c1-a717-4087-be06-d40d2070ad52:0 X-MS-Exchange-SkipListedInternetSender: ip=[209.85.218.49];domain=mail-ej1-f49.google.com X-MS-Exchange-ExternalOriginalInternetSender: ip=[209.85.218.49];domain=mail-ej1-f49.google.com X-MS-PublicTrafficType: Email X-MS-TrafficTypeDiagnostic: ML1PEPF0000F177:EE_|SY4PR01MB7171:EE_|MEYPR01MB6389:EE_|SY4PR01MB8118:EE_ X-MS-Office365-Filtering-Correlation-Id: 4f4a66d4-9301-4856-c537-08dd3d333114 X-Moderation-Data: 1/25/2025 8:34:36 PM X-LD-Processed: 82c514c1-a717-4087-be06-d40d2070ad52,ExtAddr,ExtAddr X-Auto-Response-Suppress: DR, OOF, AutoReply X-MS-Exchange-CrossTenant-Network-Message-Id: 4f4a66d4-9301-4856-c537-08dd3d333114 X-MS-Exchange-CrossTenant-Id: 82c514c1-a717-4087-be06-d40d2070ad52 X-MS-Exchange-CrossTenant-AuthSource: ML1PEPF0000F177.ausprd01.prod.outlook.com X-MS-Exchange-CrossTenant-AuthAs: Anonymous X-MS-Exchange-CrossTenant-FromEntityHeader: Internet X-MS-Exchange-CrossTenant-OriginalArrivalTime: 25 Jan 2025 20:34:38.9641 (UTC) X-MS-Exchange-Transport-CrossTenantHeadersStamped: SY4PR01MB8118 X-MS-Exchange-UnifiedGroup-DisplayName: Categories mailing list X-MS-Exchange-UnifiedGroup-Address: categories@mq.edu.au X-MS-Exchange-UnifiedGroup-MailboxGuid: 9c2f954e-92a7-451b-b723-a07075d7adb5 X-MS-Exchange-UnifiedGroup-CustomizedMessage: RedeemedBusinessGuests X-MS-Exchange-Parent-Message-Id: Auto-Submitted: auto-generated X-MS-Exchange-Generated-Message-Source: Throttled Fork Delivery Agent X-OriginatorOrg: mq.edu.au --000000000000f18555062c861e0a Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable Hi, Towards an understanding of the phenomenon why taking ultrafilters carry ov= er binary operations and associativity but not commutativity (in the sense = that if S is a semigroup then it induces a semigroup structure on =CE=B2S),= first I succeeded to generalize this to all monads over Set - I guess it = should be well known as well, but then realized that actually the unit of t= he monad is not required in the proof, so actually all 'semigroupads' (or '= semimonads') carries over semigroups. The unit of the monad is required exactly to carry over monoids. I played around with this idea, and could also come up with correspondent i= dentities involving the Kleisli star in place of the binary monad operation= . In particular, a magmad on a category C is just an endofunctor T:C->C a= nd a nat.transformation =CE=BC:TT->T (without any further requirements). For example, a 'central element' of (T,=CE=BC) would be a nat.transf. =CE= =B6:1->T satisfying =CE=BC =CE=B6T =3D =CE=BC T=CE=B6, and in this case any= magma S with central element c, we'll have =CE=B6(c) central in the induce= d magma TS. So, I basically found some answer to the question in the title: the ultrafi= lter monad itself is not commutative (whatever it should mean(?)) and there= fore commutativity is not carried over. Does anyone know about these kinds of ideas appearing in the literature? Thanks in advance, Bertalan P=C3=A9csi You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --000000000000f18555062c861e0a Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
Hi,

Towards an understanding of the phenomenon why taking ultrafilters car= ry over binary operations and associativity but not commutativity (in the s= ense that if S is a semigroup then it induces a semigroup structure on =CE= =B2S), first I succeeded to generalize this to all monads over Set  - I guess it should be well known= as well, but then realized that actually the unit of the monad is not required in the proof, so actually all = 'semigroupads' (or 'semimonads') carries over semigroups.
The unit of the monad is required exactly to carry over monoids.

I played around with this idea, and could also come up with correspondent i= dentities involving the Kleisli star in place of the binary monad operation= .
In particular, a magmad on a category  C  is just an = endofunctor  T:C->C  and a nat.transformation  =CE=BC:TT-= >T  (without any further requirements).
For example, a 'central element' of (T,=CE=BC) would be a nat.transf. = =CE=B6:1->T satisfying =CE=BC =CE=B6T =3D =CE=BC T=CE=B6, and in th= is case any magma S with central element c, we'll have =CE=B6(c) central in= the induced magma TS.

So, I basically found some answer to the question in the title: the ultrafi= lter monad itself is not commutative (whatever it should mean(?)) and therefore comm= utativity is not carried over.

Does anyone know about these kinds of ideas appearing in the literatur= e?

Thanks in advance,
   Bertalan P=C3=A9csi
 
 
You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
View group files   |   Leav= e group   |&n= bsp;  Learn more about Microsoft 365 Groups
 
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