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T Murrills (Apr 04 2020 at 00:20):oo, @Nicholas Scheel , I’ve thought about these a bit too! If I remember what I worked out a little while back it’s possible to characterize the interactions between sets of things that monoid elements act like identities towards, and the sets of things which they absorb...or something :P actually, I’m not sure I ever finished dealing with them, now that I think about it! :D
T Murrills (Apr 04 2020 at 00:23):(in the case where you consider the almost-identities (that is, they [either absorb or are absorbed by] everything) in a commutative monoid, I’m preeeeetty sure they form a total order, right? as in your intuition of “layers”... is there something analogous for noncommutative monoids, I wonder?)
Verity Scheel (Apr 04 2020 at 00:44):Can we prove it? Let a and b be two distinct almost-identities in a commutative (sub)monoid. Then either a absorbs b (a > b), in which case ab = a, or a is absorbed by b (a < b), in which case ab = b (trichotomy). Due to commutativity, if a absorbs b, then b is absorbed by a, and vice-versa, so it is anti-symmetric or something. Just need transitivity … a < b < c so ab = b and bc = c so ac = a(bc) = (ab)c = bc = c. Looks legit to me :)
Verity Scheel (Apr 04 2020 at 00:50):I'm not sure how we would define almost-identities in the non-commutative case. If they have to treat all elements equally, then they might as well be in the center of the monoid, so we can use the original definition. I'm not sure otherwise, thoughts?
T Murrills (Apr 04 2020 at 00:55):mm, true!! Maybe it would be interesting to separate the sets of “left-absorbed” and “right-absorbed” elements? (So, it would still absorb or be absorbed by everything—but whether it’s absorbed by or absorbs a given element is allowed to depend on which side we compose it on!)
T Murrills (Apr 04 2020 at 00:56):I’m not sure if that’s consistent with associativity, though!
T Murrills (Apr 04 2020 at 01:00):Hmm, it’s only allowed to vary from side to side for idempotent elements, at most, I think. Consider b an almost-identity varying from side to side for a: a(ba) = ab = a, (ab)a = aa
Verity Scheel (Apr 04 2020 at 01:09):Oh that's really interesting, so if there exists a non-idempotent element, then it cannot vary from side to side?
Verity Scheel (Apr 04 2020 at 01:11):@T Murrills thought I would move our discussion to a new thread. For reference, this is what I originally said:
I was just thinking the other day that a lot of monoids have layers of elements that are identities or “almost-identities”, and also absorbing elements or almost-absorbing elements. E.g. m <> e = m (identity), and m <> e1 = m for all m != e (almost identity).
Verity Scheel (Apr 04 2020 at 01:25):I think the case of an idempotent monoid is potentially still interesting, though, especially because a monoid generated from just these elements is idempotent.
T Murrills (Apr 04 2020 at 01:28):
T Murrills (Apr 04 2020 at 01:28):yeah! So, I scrolled allll the way back in my tumblr drafts for my math blog to find this, and here’s the part of it I didn’t remember (looks like I didn’t get the idempotent bit, though!)
at the very least it suggests a notation!
T Murrills (Apr 04 2020 at 01:29):
T Murrills (Apr 04 2020 at 01:30):(I called them “pseudoidentities”)
T Murrills (Apr 04 2020 at 01:37):So like. We have this weird “double poset” structure? Like, there’s one poset for =», and one poset for =«. And certain interactions between their upsets/downsets that I didn’t get around to figuring out! We can define these relations on the whole monoid as they’re essentially “absorbs on the right” or “absorbs on the left”. Almost-identities are the ones that act as “bottlenecks” for both partial orders: everything is either above or below them
T Murrills (Apr 04 2020 at 01:38):(I don’t like this notation a whole lot, so feel free to suggest another 😅)
T Murrills (Apr 04 2020 at 01:52):Wait. That’s not a poset...
Morgan Rogers (he/him) (Apr 04 2020 at 09:46):T Murrills said:
Hmm, it’s only allowed to vary from side to side for idempotent elements, at most, I think. Consider b an almost-identity varying from side to side for a:
a(ba) = ab = a,(ab)a = aa
The definition you previously gave already forces "almost identities" to be idempotents :slight_smile:
Morgan Rogers (he/him) (Apr 04 2020 at 09:50):Every idempotent defines a left, right and two-sided ideal; these are precisely the elements that absorb on the right, left and both sides respectively.
Verity Scheel (Apr 04 2020 at 21:10):Morgan Rogers said:
The definition you previously gave already forces "almost identities" to be idempotents :slight_smile:
Right, we were talking about a general a : M, not just the almost-identity elements.
Verity Scheel (Apr 04 2020 at 21:11):One of the other streams brought up meadows, and it turns out this same kind of idempotent structure shows up and is useful in that setting: https://arxiv.org/pdf/0903.1196v1.pdf (section 3)
Fabrizio Genovese (Apr 04 2020 at 21:24):Lol, I literally had this open in a tab right now :heart:
Fabrizio Genovese (Apr 04 2020 at 21:31):I don't understand why their base ring is taken to be commutative tho. I guess they need it in the proofs? So I guess skew-fields are not meadows?
Verity Scheel (Apr 04 2020 at 22:22):Right before the conclusion they say “every finite skew meadow is commutative”, which is a pretty nice result.
Fabrizio Genovese (Apr 04 2020 at 22:28):This seems a generalization of Wedderburn theorem, but it's strange cose in def. 2.1 meadows are assumed to be always commutative
Fabrizio Genovese (Apr 04 2020 at 22:28):Oh, now I see, they define skew-meadows in section 4
Morgan Rogers (he/him) (Apr 05 2020 at 09:43):Meadows seem like a nicer class of rings to work with than fields, cool!
Their Definition 3.11 really grates on me, though; it makes it pretty obvious that they should have chosen a different name for the complementary elements than inverses...
Nicholas Scheel said:
One of the other streams brought up meadows, and it turns out this same kind of idempotent structure shows up and is useful in that setting: https://arxiv.org/pdf/0903.1196v1.pdf (section 3)
re this, were you talking about the fact that we can put an order structure on the idempotents in a commutative monoid?