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Stream: learning: reading & references

Topic: Reading group! topics in semigroup theory

Eric M Downes (Mar 12 2024 at 14:28):

Greetings. Morgan Rogers and I are exploring a reading group concerning the algebraic aspects of monoids and semigroups.

We were thinking of meeting mondays AM US time / PM EU time, every two weeks through June, and then re-evaluating. If you're real keen on the subject but that arrangement doesn't work, please speak up.

We have not picked an initial text yet. Here are three promising resources that might serve.

Monoids, Acts, and Categories by Kilp, Knauer, and Mikhalev
(out of print and expensive; here's a link
https://www.dropbox.com/scl/fi/ulg4792f71ul5jnhvq7jo/Monoids-Acts-and-Categories.pdf?rlkey=6mc3fyhmc84gchw3pdyldwdyh&dl=0 )

Representation Theory of Finite Monoids by Steinberg
https://www.amazon.com/Representation-Theory-Finite-Monoids-Universitext-ebook/dp/B01NAITC7D/

and this lost gem by Sammy Eilenberg of CT fame
Automata Languages and Machines (somewhat more topological in its approach)
https://www.amazon.com/Automata-languages-machines-Applied-Mathematics/dp/0122340019/
(I have found a digital copy of this as well, let me know if your library doesn't have it)

Morgan's job will be to say smart insightful things, and my job will be to ask dumb questions and make mistakes. :)

Eric M Downes (Mar 14 2024 at 14:17):

After reviewing them, I vote we start with Monoids Acts & Categories. Take a look and chime in, y'all.

The stuff I really want, about monoid actions, is in III-V, but we should start at the beginning because they define a lot of terminology some of it a bit obscure/not-modern... which I don't love... I think keeping track of a glossary in notion or somesuch will be necessary.

If y'all use notion (wysiwyg LaTeX for teams with databases, etc.) I can invite you to one for personal & group notes, once we pick a book. Or we could live blog the book over at localcharts :)

Morgan Rogers (he/him) (Mar 14 2024 at 16:18):

I don't use either of those things, but the latter seems like it would work even with just one of us doing it (although it would create an imbalance in the workload :wink:)
That said, having some notes to turn into lectures in the future does sound potentially useful to me (although only potentially, since for the time being I am obliged to teach computer science)...

Eric M Downes (Mar 14 2024 at 18:42):

Oh! I have an idea for a nice book covering the usual abstract algebra subjects for computer scientists using actions and numpy... turns out array broadcasting makes numpy array expression line up very nicely with compositions, and you can do some nice things with concrete finite groups etc. that are also categorical-style. I'm using this library for research as well, and will interface to GAP for groups that are too large:
https://github.com/eric-downes/monoids

Why dont you make a notion account?
https://www.notion.so/
If you hate it we'll use something else. You can export to PDF though it doesn't always look 100% the same, but it turns out you can publish notion pages as a blog, so it may cover the other use cases as well. It's pretty easy to use unless you're given to being impatient/grumpy with software. I really enjoy having access to mathmode, markdown, and basic formatting features in one UI.

Eric M Downes (Mar 16 2024 at 00:37):

Here is monday's meeting, tentatively scheduled 1430 GMT;
https://us06web.zoom.us/j/86572493970?pwd=rbIK5g36zNtH9SBtkbzsWQBbb7rOFW.1

Morgan Rogers (he/him) (Mar 18 2024 at 14:16):

That's in 15 minutes, for anyone that wants to join!

Morgan Rogers (he/him) (Mar 18 2024 at 16:21):

In case anyone sees this late and wants to join in the next session on April 1st, we are going to independently read through all of the definitions in Chapter 1 (and maybe Chapter 2 if you're feeling ambitious) of Monoids Acts & Categories which Eric links to above so that next time we can discuss the notions which were new to us or interesting or confusing.

Theophilos (Teo) Tsantilas (Mar 18 2024 at 21:05):

Eric M Downes said:

Here is monday's meeting, tentatively scheduled 1430 GMT;
https://us06web.zoom.us/j/86572493970?pwd=rbIK5g36zNtH9SBtkbzsWQBbb7rOFW.1

Hello all! I would very much like to join the reading group if there is a place for me left... If that's the case, should I join with the link provided through zoom? Thanks!! :)

Eric M Downes (Mar 18 2024 at 21:06):

there;s definitely a place! DM me an email and I'll add you to the google calendar event

Eric M Downes (Mar 18 2024 at 21:06):

we meet every 2 weeks currently

Eric M Downes (Mar 18 2024 at 21:07):

(1430 gmt was ~6 hrs ago)

Eric M Downes (Mar 24 2024 at 14:58):

The classifying Space of a Monoid phd thesis
https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/lenzmaster.pdf

Eric M Downes (Mar 24 2024 at 14:58):

(looks interesting)

Eric M Downes (Mar 25 2024 at 07:01):

Conjecture: This diagram is sufficient to express a semigroup object in a category in which not all products exist, $\mathsf{NoProd}$ when $ж\circ ж= id_S$
proposed semigroup object dia

My reasoning is summarized by these string diagrams and "substituting" $S=M\times M\times M$ where $\times$ is an "internal" product.
sketch of a proof

The problems are:

1. I know string diagrams live in a symmetric monoidal category, which doesn't seem hopeful I admit, as these have a canonical product... and there's no guarantee that a functor out of an SMC should respect calculations of the parts as it would not generally respect the product...
2. The strings I provided imply that a suitably defined projector $\pi_i:S\to M$ would have the universal property of the product for this object, were $M$ an object of $\mathsf{NoProd}$ suggesting it cannot be if products are forbidden, or becomes trivial if products are simply not gauranteed.

Eric M Downes (Mar 25 2024 at 07:05):

(I don't know how pure we want to be in terms of topics, if anyone would prefer I just post this as a learning question elsewhere, please speak up!)

Morgan Rogers (he/him) (Mar 25 2024 at 09:52):

I saw that you mentioned it elsewhere, but I haven't looked at it closely yet. If you don't get any responses it's something we can discuss in the next reading session.

Eric M Downes (Apr 01 2024 at 10:44):

Just a reminder to everyone that we're meeting today, in... ~4 hours!
https://us06web.zoom.us/j/86572493970?pwd=rbIK5g36zNtH9SBtkbzsWQBbb7rOFW.1

Julius Hamilton (Apr 01 2024 at 10:53):

I would want to but I think I’m just not quite ready yet. It still takes me a long time to mull over new mathematical ideas. How many people are in the group?

Morgan Rogers (he/him) (Apr 01 2024 at 13:06):

So in... 85 mins?

Morgan Rogers (he/him) (Apr 01 2024 at 13:06):

It was just two of us last time.

Julius Hamilton (Apr 01 2024 at 13:31):

I think I’ll try it out to see how it is.

Morgan Rogers (he/him) (Apr 01 2024 at 13:51):

Be sure to take a look at the first chapter of the textbook before hand, because I imagine we will be skipping through it in the discussion :)

Julius Hamilton (Apr 01 2024 at 14:07):

I’ll try. I believe it starts at 8:30am MT = 14:30 UTC?
Is it a one hour meeting?

Julius Hamilton (Apr 01 2024 at 14:07):

Thank you

Morgan Rogers (he/him) (Apr 01 2024 at 14:11):

Yes, UTC lines up with GMT now ;)

Eric M Downes (Apr 01 2024 at 14:27):

In case we need a whiteboard; https://scribbletogether.com/whiteboard/7AAC7A35-8F56-48CB-AF3B-5F7D83B0A8B7

Julius Hamilton (Apr 01 2024 at 14:28):

Cool. I will not be able to follow but I will listen in to get motivated to study at a higher level.

Julius Hamilton (Apr 01 2024 at 14:56):

Accidentally left meeting and would like to rejoin

Eric M Downes (Apr 01 2024 at 15:56):

Ah sorry about that, I will try to make it so people dont need my permission to join; zoom makes that annoying

Eric M Downes (Apr 01 2024 at 15:57):

I'll also add you to the gcal event

Julius Hamilton (Apr 01 2024 at 16:00):

All good.
It seemed like I could learn a lot from those meetings so definitely would love to join the next one. Thanks.

Eric M Downes (Apr 12 2024 at 02:53):

Interesting discussion of semigroup structure theorems I ran into today.
https://mathoverflow.net/questions/131110/what-are-the-main-structure-theorems-on-finitely-generated-commutative-monoids

see everybody monday!

Naso (Apr 12 2024 at 02:59):

I'm really interested in learning more about semigroups (and other semi- stuff: semirings, semimodules, ...), but I'm not sure if the time will suit me (I'm in Australia, UTC+10) :smiling_face_with_tear:

Are you guys writing up some notes?

Eric M Downes (Apr 12 2024 at 03:02):

we can discuss changing the time, although I agree there isnt much overlap. I'll invite you to the notion where we are collecting notes if you dm me your email address. (Notion is mostly markdown but critically it has mathmode "$$", so good for notes.

Julius Hamilton (Apr 13 2024 at 21:35):

Is there official reading for Monday’s meeting?

Julius Hamilton (Apr 13 2024 at 21:48):

This looks sort of interesting and relevant - https://en.m.wikipedia.org/wiki/Light%27s_associativity_test

Julius Hamilton (Apr 13 2024 at 21:53):

Eric M Downes said:

Conjecture: This diagram is sufficient to express a semigroup object in a category in which not all products exist, $\mathsf{NoProd}$ when $ж\circ ж= id_S$
proposed semigroup object dia

My reasoning is summarized by these string diagrams and "substituting" $S=M\times M\times M$ where $\times$ is an "internal" product.
sketch of a proof

The problems are:

1. I know string diagrams live in a symmetric monoidal category, which doesn't seem hopeful I admit, as these have a canonical product... and there's no guarantee that a functor out of an SMC should respect calculations of the parts as it would not generally respect the product...
2. The strings I provided imply that a suitably defined projector $\pi_i:S\to M$ would have the universal property of the product for this object, were $M$ an object of $\mathsf{NoProd}$ suggesting it cannot be if products are forbidden, or becomes trivial if products are simply not gauranteed.

I’d like to understand this more. Could you explain what those string diagrams are saying?

Eric M Downes (Apr 14 2024 at 00:43):

Official reading is "make more progress on ch 1 of KKM", linked far above.

Re associativity tests, for finite magmas I prefer using action-composition; as soon as you find a composition of two rows (or columns) not already a row (column) you know the magma is not associative, and you stop.

The string diagrams are an internalization of the operations from the lawvere theory; take a look at Ch 1 of MacLane for a monoid object. So instead of the usual $\mu:M^2\to M$ I am writing $\mu:M^3\to M^3$ but I'm calling $M^3$ its own object. The "o-" symbol in the string dias is a monadic unit that introduces from nothing a default value (so probably I should just make it a monoid and return the monoidal identity):

$\mu:(x,y,z)\mapsto(xy,z,1)$

Eric M Downes (Apr 14 2024 at 00:45):

and ж is just a transposition of arguments.

mind you these are not how it always needs to work, these are the localization to show my diagram includes the traditional semigroup object. Its quite possible thats the only real case that can exist.

Morgan Rogers (he/him) (Apr 15 2024 at 14:00):

Reminder that this will happen in 30mins

Morgan Rogers (he/him) (Apr 15 2024 at 14:32):

@Eric M Downes same link from last time?

Morgan Rogers (he/him) (Apr 15 2024 at 14:40):

Hmm... I suspect that either the host has forgotten or I have made a timing error.

Julius Hamilton (Apr 15 2024 at 14:43):

I had the same time on my calendar

Julius Hamilton (Apr 15 2024 at 14:43):

I was just commuting

Julius Hamilton (Apr 15 2024 at 14:43):

Wasn’t sure if I would join

Morgan Rogers (he/him) (Apr 15 2024 at 14:44):

Julius Hamilton said:

I was just commuting

JH = HJ?
You can join the meeting with the Zoom link further up in the chat now :)

Julius Hamilton (Apr 15 2024 at 14:45):

Haha nice!!!

Julius Hamilton (Apr 15 2024 at 14:46):

I will join via chat cos I’m in a public environment

Julius Hamilton (Apr 15 2024 at 14:49):

Eric said we would change the settings so we don’t have to be “let in”, would be good to do that today

Julius Hamilton (Apr 15 2024 at 14:49):

E0B82C98-BB6C-4024-B61D-38DAFCFF5C48.png

Eric M Downes (Apr 15 2024 at 15:55):

Sadly I did change the settings and zoom still wont do it :(... you might need to be logged into to the zoom account for the same email as in the google invite? I'll look into it.

We got a late start today, but made it up to Green's relations.

My biggest piece of homework is to show that the factor semigroup of the rees congruence is a pushout in $\mathsf{Set}^M$ the category of $M$-actions.

Eric M Downes (Apr 15 2024 at 16:31):

Ahhhhhhh I think I figured it out; sorry guys. Appears I had to both

• turn ON "Allow participants to join before host"
• AND turn OFF "Enable waiting room" (which I had not done)

so, hopefully that will work? :/

Theophilos Tsantilas (Apr 16 2024 at 09:39):

Unfortunately I wasn't able to attend yesterday. Which topics did you cover?

Eric M Downes (Apr 16 2024 at 11:11):

at this point we covered everything in Ch 1 up to green's relations.

Eric M Downes (Apr 25 2024 at 16:55):

I had a fun time writing the Rees congreunce itself as a semigroup homomorphism. Thought I'd share.

Let's say you have a semigroup object $(M,\mu)$ in a topos in which the subobject classifier is a meet semilattice (idempotent monoid) $(\Omega,\land,\top)$ where as usual $\top:1\to\Omega$, and
$\top_M=\top\circ\,!:M\to\Omega$ is pre-composition with the unique terminal map $!:M\to 1$.

Then you can rewrite congruence in an appealing way. Here $\overset{\alpha}{\implies}$ is a not-necessarily invertible 2-morphism (which acts as $\bigwedge$ on the codomain of all its argument morphisms), $\Delta$ doubles the argument, and $(12),(23)$ are transpositions of arguments:

$\sim\,:(M,\mu)\times (M,\mu)\to (\Omega,\land,\top)$ is a congruence just when it is

• commutative $\sim\circ\,(12)=\,\sim$
• reflexive (~idempotent in image) $\sim\circ\,\Delta=\top_M$
• transitive $\land\circ(\sim\times\sim)\circ(id_M\times\Delta\times id_M)\overset{\alpha}{\implies}\land\circ\,(\sim\times\top_M)\circ(23)$
• congruent (one-way homomorphic) $\land\circ(\sim\times\sim)\overset{\beta}{\implies}\sim\circ\ (\mu\times\mu)$

I think this is pretty cool, though I'm unsure how seriously to take it.

Eric M Downes (Apr 25 2024 at 17:22):

The transitivity rule reminds me of how $\infty$-categoricists talk about composition being associative, not up to equality, but up to a higher morphism that witnesses the composition. I lack the understanding to really take it further than that, but it suggests this picture views categories themselves as "higher" Lawvere theories (in which strict equality is not required among the equations) ... or something. Probably understanding if this picture is consistent with [[congruence]] would be fruitful.

Ok, have a nice day!

Morgan Rogers (he/him) (Apr 25 2024 at 18:57):

That sounds valid for a general congruence. Where does the Rees congruence come in?

Eric M Downes (Apr 25 2024 at 19:05):

Glad it passes a minimal sniff-test! Thanks, and fair question. I guess I didn't write that part! I think (my translation of a) Rees congruence would be any morphism with the same signature as $\sim$ with a monic 2-morph into $\sim$.

Morgan Rogers (he/him) (Apr 28 2024 at 10:28):

Hi folks, I exceptionally won't be able to make it tomorrow. Would it be okay to move it back a week?

Eric M Downes (Apr 28 2024 at 12:42):

Fine by me -- unless there are objections, we'll meet two weeks in a row: next week, and the week after. That way if anyone had planned around the 2-week cycle it won't be disrupted.

I would also consider having a discussion Mon USA PM / Tue AUS AM. @Naso let me know if that's useful.

Eric M Downes (May 06 2024 at 13:51):

We're meeting in 40 minutes!
https://us06web.zoom.us/j/86572493970?pwd=rbIK5g36zNtH9SBtkbzsWQBbb7rOFW.1

Eric M Downes (May 07 2024 at 21:42):

Re-posting a paper brought to my attention by @Brendan Murphy
https://www.cse.chalmers.se/~peterd/papers/Coherence_Monoidal.pdf

What other theorems about monoids lift to monoidal categories this way? What do actions look like?

Eric M Downes (May 13 2024 at 14:49):

Hi all, reminder we will next meet on the 27th to cover $\S$ 1.4+. No meeting next week, kthxbyeeee.

Morgan Rogers (he/him) (May 13 2024 at 19:45):

I was rudely absent this afternoon when the meeting was supposed to take place, I apologize.

Eric M Downes (May 27 2024 at 10:49):

See everyone in a few hours!

Morgan Rogers (he/him) (May 27 2024 at 13:20):

In ~70 mins, right?

Morgan Rogers (he/him) (May 27 2024 at 15:38):

For anyone who cares, we stopped just before 4.20 in Chapter 1 this session.

Eric M Downes (May 29 2024 at 06:41):

During our last session, IIRC Morgan wanted to know more about example 4.9.10, which I happen to have experience with. I figure I'd summarize a few facts about it and Morgan can challenge anything he wants more detail on or doesn't believe. I'l close by rambling incoherently, as I am wont to do.

We have a monoid $\mu:MM\to M$,
-- so its direct image $\mu^\vdash:2^M2^M\to2^M$ is also a monoid.
-- this makes $2^M$ into a residuated lattice; $\mu$ distributes over $\cap,\cup$.

... acting on $A$ via a curried action $\alpha:M\to End(A)$.

• so $2^{End(A)}$ is a residuated lattice using function composition as the product.
• with a closure operator on both ends taking a subset to the monoid it generates
• all magma-actions** can be realized as maps $a: X\to A^A$ from possibly not-moore-closed $X\subseteq M$.
• the moore closure on both sides then gives the enveloping monoid action (adjoint monoid in Lie Algebra); the smallest associative monoid with the magma as a retract.

** -- that is, not necessarily associative maps $XA\to A$.

Now time for the crazy-person ranting you have all come to expect from my communications.

The general functor-category in which the magma action lives in that last point, expressing its closure, would be $\sf [discrete(X), X^*] ⨾ [X^*, cl\circ a^\vdash(X)]$, where the final target category is the enveloping monoid, and the intermediate monoid is the Kleene-star, and the second functor is the unique monoid congruence. The discrete category needs in the general case, at least one non-identity morphism per generator. The composition data is then specified in the fibration.

This is sort of formal and non-constructive in the general case. In contrast, the powerset picture is constructive, and for magmas of order $\lesssim$ 1000, even computable on a decade-old computer. :)

If you have a magma with more structure, you can be less abstract and have the source morphisms carry more of the weight; I think for instance $discrete(\mathbb{O})$ should be the discrete groupoid containing three copies of the quaternions $\mathbb{H}$.

But even in the non-constructive case, this also suggests interesting intermediate concepts of closures for magmas. Like, maybe you don't immediately project down every object to a single object... you could have a whole series of intermediate steps to categories with fewer and fewer objects. I believe this series of fibrations just recapitulates the ascending chains of the submonoid lattice that contain the magma. But where Morgan might be able to see if this is generally true is working with functors between topoi, so instead of a residuated powerset lattice, we have presheaves to the subobject classifier.

Eric M Downes (Jun 10 2024 at 13:11):

This is your 80 minute warning for The Semigroup Variety Hour (starring Morgan Rogers), said to be America's most highly-rated semigroup zoom content.** So full of actions, you'll forget your identity!

** -- the ratings in Europe are undefined due to the appearance of a conflict of interest.

Morgan Rogers (he/him) (Jun 24 2024 at 15:04):

I apologize: I am at CT this week and will be at conferences for the next two Mondays too..! Maybe we could have a final pre-holiday session in 3 weeks?

Eric M Downes (Jun 24 2024 at 16:38):

Done! See you Jul 15; I don't know when your holiday ends -- should we get in touch early Sept to schedule going forward?

Morgan Rogers (he/him) (Jun 24 2024 at 16:46):

Sure. Maybe some people will join at that point. Nothing we've done is too technical, I'm sure they could join in at the point we're at as long as we're there to fill them in on the outdated conventions in this textbook hahaha

Eric M Downes (Jun 24 2024 at 17:05):

To be clear: Even if they don't, I intend to soldier on! Both of us want to learn the material, oudated conventions be damned, so we may as well do it with some fraternité et solidarité, etc. (or should that only apply to studying semigroups that are also isomorphic and free?)

Morgan Rogers (he/him) (Jul 15 2024 at 14:49):

Wait did I miss the start time somehow?

Morgan Rogers (he/him) (Jul 15 2024 at 14:50):

I just got an automatically generated email from @Julius Hamilton 's bot :')

Morgan Rogers (he/him) (Jul 15 2024 at 14:52):

I guess I did... oops :exhausted:

Julius Hamilton (Jul 15 2024 at 14:53):

I’m afraid I don’t know of any bot I have.

Julius Hamilton (Jul 15 2024 at 14:53):

It may be that I RSVPed to a Google Calendar event or something though.

Morgan Rogers (he/him) (Jul 15 2024 at 14:54):

Hahaha Fireflies has been attending our meetings on your behalf and claims to have been sent by you

Julius Hamilton (Jul 15 2024 at 14:54):

Weird! Time to clean up my email inbox I guess :smile:

Julius Hamilton (Jul 15 2024 at 14:55):

That said I would like to join the next meeting. When is it?

Morgan Rogers (he/him) (Jul 15 2024 at 14:55):

It was supposed to be today, but I missed the start time by about 20 mins :grimacing:

Julius Hamilton (Jul 15 2024 at 14:56):

Ok well I’m happy to reschedule now that we’re both aware of this.

Morgan Rogers (he/him) (Jul 15 2024 at 14:56):

@Eric M Downes you can pick the time, since I feel I've messed you around rather...

Eric M Downes (Jul 15 2024 at 15:06):

What we agreed last time was to revisit this in September when Morgan’s teaching schedule was known, and after his vacation.

Today was a last hurrah before a long break anyway, so why don’t we just start in Sept with a review, and go from there.

I’ll message this topic around the first week of September, and we’ll figure out schedules.

Eric M Downes (Jul 15 2024 at 15:12):

And yes, Fireflies claimed to have been sent by Julius. We never actually checked but Monoids are commonly used to model automata, so it felt especially discriminatory to exclude it. Plus it was very quiet and well-behaved; only asked once or twice about the whereabouts of someone named Sarah Conor. shrug

Julius Hamilton (Jul 15 2024 at 15:23):

Monoids are used to model automata so it felt discriminatory to exclude it

Lol, I see what you did there! :wink: