I see a pentagon dimple.

Yep! Couple a beers but no AI involved.

BTW same result and proof holds if we assume all spaces are compact and weakly Hausdorff.

Non-trivial example: Alexandroff compactifcation X of your favorite T_2 space Y which is NOT locally compact , e.g. Y is rationals with order topology.

Catchy rhyme, Say it in time.

Useful fact, Closed is compact.

Problem is hacked, Push forward and back.

Yeah, I'm waffling a bit too.

If the starting monoid is cancellative, abelian, totally ordered, and if all nonempty's have a greatest lower bound, for example if M is the natural numbers, then x-->[0,x] is the way to go, mapping nonzero x-->[0,x) is NOT a homomorphism, even if we map 0 to {0}.

But we can in fact get best of both of worlds employing your ideas!

Map 0 to {0} and nonzero x to [0,x). The submonoid K is the bounded sets in 2^M of the form {0} union any bounded union of half open intervals [0,x).

K is isomorphic to additive non-negs, and M-->K is a monoid monomorphism.

Yes! Thanks! Indeed one stop shopping as colimit, the additive naturals or integers, indexed over naturals, with monomorphic bonding maps x--> N(n) x with N(n) = (2 3 5 ...p(n))^n.

(The less elegant inclusions of fractions is useful as a sanity check too).

But don't we get additive rationals as colimit of infinite cyclic

groups A1 -->A2-->A3... with inclusion bonding maps,

A_n the rational fractions m/[(2 3 5 ...p_n)^n] ?

(m is an integer and p_n the nth prime)

Arghh! Thanks!

Replace with the additive dyadic rationals,

the colimit of the infinite

cyclic group Z under the doubling map

x-->x+x, indexed over the naturals

Z-->Z-->Z...

A monoids versus semigroups categorical tussle, MUST identities map to identities under maps which preserve multiplication?

Good ways to construct additive reals avoiding Dedekind cuts and Cauchy sequences in short supply.

In this case, (at most) 2-1 quotients are compatible with the algebra.

Yes! But the open ray [0,0) can be uninhabited?

So if we want x-->[0,x] to be a monomorphism, M--> 2^M, (adding subsets A and B of M, to get A+B in the codomain) we should to KEEP the closed intervals, so that 0 maps to [0,0].

i.e. the formula x-->[0,x) breaks when x=0, and M is skeletal.

Another Saturday night and I aint got no monoids.

The late great

Say I'm co kernel.

That's the sound o the chain ...homoTop...

Don't know much about histograms....

Eliahu Levy (1947-2023) I was shocked and saddened last week to hear that my friend and colleague Eliahu Levy passed away. Being a research mathematician, I have had the good fortune of meeting some very brilliant individ…

Orr Shalit has a nice memorial post

Eliahu Levy (1947-2023)

noncommutativeanalysis.wordpress.com/2023/08/09/e...

Can the right divisibility preorder on a noncommutative monoid be total?

(a<=b if ac=b for some c).

Yes, according to Eliahu Levy ( 1947-2023).

arxiv.org/pdf/2006.00886

Definitely.

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Thanks!

I am having my come to semi-groups moment.

Thanks Michael,

Indeed, I should have used preorder instead, I was conflating, (incorrectly?) preorder with nonstrict poset.

but preorder relation is the way to go x<=x, plus transitivity.

Thanks!

Indeed the relation is transitive and that's about it.

I am allowing for nonstrict posets,

e.g. if the starting monoid is a group we have x<=y universally.

(I will delete, edit, or clarify if I've posted obvious nonsense).

If the abeliain monoid (M,*) determines a total order on (M,<), then the bounded rays comprise a totally ordered and left complete submonoid of 2^M.

Now take the cancellative quotient, and we get the non-neg additive reals, if M is the non-neg dyadic rationals.

All this is `easy'.

f:M-->G is trivial if there is k in M so that km=mk=k for all m in M,

if M is a monoid and G is a group, and f is a homomorphism.

Ignoring order, the additive rationals are discrete, a colimit of cyclic groups.

But if we restrict to the monoid of non-neg rationals, algebra gives us x<y iff x+z=y, we get order topology.

Induced order on monoid of bounded rays, almost the non-neg reals. Mod out by [0,x)=[0,x] when x rational.

Submonoid category. Exercises and examples.

From S={a,b} we can create M(S), the monoid of functions

f:M-->M under composition.

From M(S) we can create the category C(S) of submonoids of M(S).

|C(S)|=5 in this case.

We can then ask, given x and y in C,

are x and y isomorphic?

|H(x,y)|=?

NEVER meaning this relation yields the trivial order on a group, unlikely useful since the starting data is inhabited.

Monoids are essential building blocks of groups.

We sometimes get a useful partial order on a monoid declaring x<=y if zx=y has a solution.

Sometimes this internal algebra creates a total order, e.g. on the natural numbers.

But this NEVER happens in a group, since x <= y holds for all x and y.

Malignant narcissists experience increased paranoia, denial, and rage as death nears. They often externalize these experiences through destruction and abuse of others in an effort to maintain a sense of control.

I suspect all of these wars are in part related to Trump’s awareness of his mortality.

War or Nothing A tale across the years; seven lessons learned from a quarter-century in a war-oriented society, where the greatest crime is any opposition to killing.

Seven lessons learned from a quarter century in a war-oriented society.

It's 2001—the year the movies promised we'd make contact with aliens—and the United States has rather recently been attacked by terrorists who flew passenger planes into buildings.

www.the-reframe.com/war-or-nothi...

Yes, despite pragmatic rarity of checking off conditions.

Metrizability demands monoid of additive non-negative reals.

Origin stories of latter highly nontrivial, done carefully.

First class examples of limits, colimits, and monoid quotients emerge, done with care.

Reusable tools writ large.

Hating the Game The cooperation game, the murder game, and acting in good faith with people you know are acting in bad faith.

Today I wrote about the racist accommodation to the racist response to Bad Bunny's Super Bowl show; the moral hazard of extending good faith to those who act in bad faith; and how alienating white bigots is actually part of the solution, not a part of the problem. www.the-reframe.com/hating-the-g...

For cancellative abelian monoids, a nontrivial quotient map can have trivial kernel.

Map the free monoid F(C) over Cantor space C onto the additive non-negative reals, so that the right-shift map of F(C) represents halving.

Knowledge of the kernel of a quotient map is NOT one stop shopping.