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| 2025-10-14 16:14:01 +0200 | <ski> | unless the key monoid is highly non-injective, i guess |
| 2025-10-14 16:13:43 +0200 | <tomsmeding> | that's not usually what you want in practice |
| 2025-10-14 16:13:42 +0200 | <ski> | right |
| 2025-10-14 16:13:30 +0200 | <tomsmeding> | yes, the other problem is that the maps get very big this way |
| 2025-10-14 16:13:25 +0200 | <ski> | wanting all combinations of the keys in one map, with the keys in the other map |
| 2025-10-14 16:12:57 +0200 | <ski> | yea .. it kinda has a "tensor feel" |
| 2025-10-14 16:12:30 +0200 | <tomsmeding> | *originals |
| 2025-10-14 16:12:21 +0200 | <tomsmeding> | feels a bit far-fetched to me, not least because the result "invents new keys" that were not there in the original |
| 2025-10-14 16:10:58 +0200 | <ski> | now .. is there a use case for wanting to multiply `Map k v's, given `Monoid k' ? |
| 2025-10-14 16:09:47 +0200 | <ski> | so you can no longer keep track of a single element per group (monoid)) |
| 2025-10-14 16:09:41 +0200 | <ski> | the direct sum case involves a function, with *finite support*, from the family indices to elements of the corresponding groups (monoids), while for direct product, it's arbitrary such functions. for arbitrary (not necessarily commutative/abelian) groups (monoids), though, the categorical coproduct case (called "free product") becomes a larger object. `g_0 * h * g_1' is no longer equal to `(g_0 * g_1) * h', |
| 2025-10-14 16:09:35 +0200 | <ski> | (oh, and when i said "formal polynomial", above, i had "formal power series in mind" .. so wondering whether that would involve a cofree, rather than free, monoid. cf. how direct sum (categorical coproduct) and direct product (categorical product) for *commutative* groups (as well as monoids) coincide, for a *finite* family of groups (or monoids), but are distinct for an infinite family. difference is that |
| 2025-10-14 16:08:42 +0200 | MelodyOwO | (~MelodyOwO@user/MelodyOwO) (Quit: Leaving.) |
| 2025-10-14 16:07:48 +0200 | inline | (~inlinE@ip-178-202-059-161.um47.pools.vodafone-ip.de) Inline |
| 2025-10-14 16:07:18 +0200 | inline | (~inlinE@ip-178-202-059-142.um47.pools.vodafone-ip.de) (Ping timeout: 252 seconds) |
| 2025-10-14 16:06:35 +0200 | <tomsmeding> | then the structure of the monoid suddenly comes into play |
| 2025-10-14 16:06:11 +0200 | <tomsmeding> | but yeah I see where my understanding went wrong: I wasn't properly thinking about the fact that this R[G] is supposed to be a _ring_, and what the multiplication operation ought to do |
| 2025-10-14 16:05:05 +0200 | <tomsmeding> | polynomials on Z/2Z have a bunch of such "redundancies" |
| 2025-10-14 16:02:37 +0200 | fp | (~Thunderbi@2001:708:20:1406::10c5) (Ping timeout: 246 seconds) |
| 2025-10-14 16:02:33 +0200 | <ski> | (iirc, in some rings, (ordinary) polynomials can be distinct (having distinct coefficients), while still having the same value at each possible input (being extensionally equal, the corresponding functions to the polynomials being equal)) |
| 2025-10-14 16:01:19 +0200 | <ski> | the "formal" here means that when we write a (finite) sum of products of monoid elements and associated coefficients, this is just a suggestive notation for having a function from the monoid elements to the coefficients, with "finite support" (meaning only finitely many monoid elements map to non-zero coefficients) |
| 2025-10-14 15:59:36 +0200 | <tomsmeding> | thanks :) |
| 2025-10-14 15:58:08 +0200 | <tomsmeding> | ski: "not-necessarily-injective" -- ah! right |
| 2025-10-14 15:56:59 +0200 | <ski> | (and yea, i was reminded of this, by the talk about `Monoid (Map k v)'. if we ignore the multiplication (and subtraction/negation) in the monoid ring, then `k' corresponds to the set of indeterminates (generators) in the "polynomials", and `v' corresponds to the monoid of coefficients) |
| 2025-10-14 15:53:47 +0200 | <ski> | instead of combining monomials like `x * y^2' and `x^3 * z' into `x^4 * y^2 * z', amounting to bag/multiset merging (multiplication in the free commutative monoid), you get a not-necessarily-injective multiplication in the monoid |
| 2025-10-14 15:51:33 +0200 | weary-traveler | (~user@user/user363627) (Remote host closed the connection) |
| 2025-10-14 15:51:30 +0200 | <ski> | "so whether the elements of G do something with each other is not relevant for how many elements rae in R[G], it seems" -- it affects multiplication of them, yea |
| 2025-10-14 15:50:20 +0200 | <ski> | "More formally, `R[G]' is the free `R'-module on the set `G', endowed with `R'-linear multiplication defined on the base elements by `g·h := gh', where the left-hand side is understood as the multiplication in `R[G]' and the right-hand side is understood in `G'." |
| 2025-10-14 15:45:50 +0200 | <tomsmeding> | (I guess it would be a "monoid module") |
| 2025-10-14 15:45:00 +0200 | ystael | (~ystael@user/ystael) ystael |
| 2025-10-14 15:44:58 +0200 | <tomsmeding> | if the sums weren't formal, this would be a module, would it not? |
| 2025-10-14 15:44:33 +0200 | <tomsmeding> | so whether the elements of G do something with each other is not relevant for how many elements rae in R[G], it seems |
| 2025-10-14 15:44:10 +0200 | <tomsmeding> | ski: well, the article does say that the polynomials are formal ("set of formal sums") |
| 2025-10-14 15:30:45 +0200 | bitdex | (~bitdex@gateway/tor-sasl/bitdex) bitdex |
| 2025-10-14 15:30:07 +0200 | bitdex | (~bitdex@gateway/tor-sasl/bitdex) (Remote host closed the connection) |
| 2025-10-14 15:28:44 +0200 | <ski> | tomsmeding : i guess saying "polynomial" implies that the monoid is the free (commutative) monoid (hm, for "formal polynomial", would that be cofree monoid ?) |
| 2025-10-14 15:25:59 +0200 | bitdex | (~bitdex@gateway/tor-sasl/bitdex) bitdex |
| 2025-10-14 15:17:53 +0200 | inline | (~inlinE@ip-178-202-059-142.um47.pools.vodafone-ip.de) Inline |
| 2025-10-14 15:17:36 +0200 | inline | (~inlinE@ip-178-202-059-161.um47.pools.vodafone-ip.de) (Ping timeout: 256 seconds) |
| 2025-10-14 15:16:13 +0200 | Googulator | (~Googulato@2a01-036d-0106-03fa-dc7a-fb6e-71bb-aaf0.pool6.digikabel.hu) (Quit: Client closed) |
| 2025-10-14 15:15:49 +0200 | Googulator6 | (~Googulato@2a01-036d-0106-03fa-dc7a-fb6e-71bb-aaf0.pool6.digikabel.hu) |
| 2025-10-14 15:14:49 +0200 | luna___ | (~luna@fedora/bittin) bittin |
| 2025-10-14 15:07:37 +0200 | CiaoSen | (~Jura@2a02:8071:64e1:da0:5a47:caff:fe78:33db) (Ping timeout: 244 seconds) |
| 2025-10-14 15:05:42 +0200 | kuribas | (~user@2a02-1810-2825-6000-b5ac-98ee-b19a-ab1f.ip6.access.telenet.be) (Ping timeout: 256 seconds) |
| 2025-10-14 15:03:45 +0200 | inline | (~inlinE@ip-178-202-059-161.um47.pools.vodafone-ip.de) Inline |
| 2025-10-14 15:00:52 +0200 | Googulator49 | Googulator |
| 2025-10-14 14:50:50 +0200 | Googulator75 | (~Googulato@2a01-036d-0106-03fa-dc7a-fb6e-71bb-aaf0.pool6.digikabel.hu) (Client Quit) |
| 2025-10-14 14:50:46 +0200 | Googulator49 | (~Googulato@2a01-036d-0106-03fa-dc7a-fb6e-71bb-aaf0.pool6.digikabel.hu) |
| 2025-10-14 14:46:14 +0200 | Googulator81 | (~Googulato@2a01-036d-0106-03fa-dc7a-fb6e-71bb-aaf0.pool6.digikabel.hu) (Quit: Client closed) |
| 2025-10-14 14:46:12 +0200 | rvalue | (~rvalue@about/hackers/rvalue) rvalue |