Author: IMC

Have you ever come across the words “commutative diagram” before? Perhaps you’ve read or heard someone utter a sentence that went something like

“For every [bla bla] there existsa [yadda yadda] such that the following diagram commutes.”

and perhaps it left you wondering what it all meant. I was certainly mystified when I first came across a commutative diagram. And it didn’t take long to realize that the phrase “the following diagram commutes” was (and is) quite ubiquitous. It appears in theorems, propositions, lemmas and corollaries almost everywhere!

So what’s the big deal with diagrams? And what does commute mean anyway?? It turns out the answer is quite simple.

Do you know what a composition of two functions is?

Then you know what a commutative diagram is!

A commutative diagram is simply the picture behind function composition.

Truly, it is that simple. To see this, suppose AA and BB are sets and ff is a function from AA to BB. Since ff maps (i.e. assigns) elements in AA to elements in BB, it is often helpful to denote that process by an arrow.

And there you go. That’s an example of a diagram. But suppose we have another function gg from sets BB to CC, and suppose ff and gg are composable. Let’s denote their composition by h=g∘fh=g∘f. Then both gg and hh can be depicted as arrows, too.

But what is the arrow A→CA→C really? I mean, really? Really it’s just the arrows ff and gg lined up side-by-side.

But maybe we think that drawing hh’s arrow curved upwards like that takes up too much space, so let’s bend the diagram a bit and redraw it like this:

This little triangle is the paradigm example of a commutative diagram. It’s a diagram because it’s a schematic picture of arrows that represent functions. And it commutes because the diagonal function IS EQUAL TO the composition of the vertical and horizontal functions, i.e. h(a)=g(f(a))h(a)=g(f(a)) for every a∈Aa∈A. So a diagram “commutes” if all paths that share a common starting and ending point are the same. In other words, your diagram commutes if it doesn’t matter how you commute from one location to another in the diagram.

But be careful.

Not every diagram is a commutative diagram.

The picture on the right is a bona fide diagram of real-valued functions, but it is defintitely not commutative. If we trace the number 11 around the diagram, it maps to 00 along the diagonal arrow, but it maps to 11 itself if we take the horizontal-then-vertical route. And 0≠10≠1. So to indicate if/when a given diagram is commutative, we have to say it explicitly. Or sometimes folks will use the symbols shown below to indicate commutativity:

‍

I think now is a good time to decode another phrase that often accompanies the commutative-diagram parlance. Returning to our f,g,hf,g,h example, we assumed that f,gf,g and h=g∘fh=g∘f already existed. But suppose we only knew about the existence of f:A→Bf:A→B and some other map, say, z:A→Cz:A→C. Then we might like to know, “Does there exist a map g:B→Cg:B→C such that z=g∘fz=g∘f? Perhaps the answer is no. Or perhaps the answer is yes, but only under certain hypotheses.* Well, if such a gg does exists, then we’ll say “…there exists a map gg such that the following diagram commutes:

but folks might also say

“…there exists a map gg such that zz factors through gg”

The word “factors” means just what you think it means. The diagram commutes if and only if z=g∘fz=g∘f, and that notation suggests that we may think of gg as a factor of zz, analogous to how 22 is a factor of 66 since 6=2⋅36=2⋅3.

By the way, we’ve only chatted about sets and functions so far, but diagrams make sense in any context in which you have mathematical objects and arrows. So we can talk about diagrams of groups and group homomorphisms, or vector spaces and linear transformations, or topological spaces and continuous maps, or smooth manifolds and smooth maps, or functors and natural transformations and so on. Diagrams make sense in any category. And as you can imagine, there are more complicated diagrams than triangular ones. For instance, suppose we have two more maps i:A→Di:A→D and j:D→Cj:D→C such that hh is equal to not only g∘fg∘f but also j∘ij∘i. Then we can express the equality g∘f=h=j∘ig∘f=h=j∘i by a square:

Again, commutativity simply tells us that the three ways of getting from AA to CC are all equivalent. And diagrams can get really crazy and involve other shapes too. They can even be three-dimensional! Here are some possibilities where I’ve used bullets in lieu of letters for the source and target of the arrows.

No matter the shape, the idea is the same: Any map can be thought of as a path or a process from AA to BB, from start to finish. And we use diagrams to capitalize on that by literally writing down “AA” and “BB” (or “∙∙” and “∙∙”) and by literally drawing a path—in the form of an arrow—between them.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

I am often asked the question, “How do you do it?!” Now while I don’t think my note-taking strategy is particularly special, I am happy to share! I’ll preface the information by stating what you probably already know: I LOVE to write.* I am a very visual learner and often need to go through the physical act of writing things down in order for information to “stick.” So while some people think aloud (or quietly),

I think on paper.

My study habits, then, are built on this fact. Of course not everyone learns in this way, so this post is not intended to be a how-to guide. It’s just a here’s-what-I-do guide.

With that said, below is a step-by-step process I tried to follow during my final years of undergrad and first two years of grad school.**

‍Step 1

Read the appropriate chapter/section in the book before class

I am an “active reader,” so my books have tons of scribbles, underlines, questions, and “aha” moments written on the pages. I like to write while I read because it gives me time to pause and think about the material. For me, reading a mathematical text is not like reading a novel. It often takes me a long time just to understand a single paragraph! Or a single sentence. I also like to mark things that I don’t understand so I’ll know what to look for in the upcoming lecture.

STEP 2

Attend lecture and take notes

This step is pretty self-explanatory, but I will mention this: I write down much more than what is written on the chalkboard (or whiteboard). In fact, a good portion of my in-class notes consists of what the professor has said but hasn’t written.

‍My arsenal

‍STEP 3

Rewrite lecture notes at home

My in-class notes are often an incomprehensible mess of frantically-scribbled hieroglyphs. So when I go home, I like to rewrite everything in a more organized fashion. This gives the information time to simmer and marinate in my brain. I’m able to ponder each statement at my own pace, fill in any gaps, and/or work through any exercises the professor might have suggested. I’ll also refer back to the textbook as needed.

Sometimes while rewriting these notes, I’ll copy things word-for-word (either from the lecture, the textbook, or both), especially if the material is very new or very dense. Although this can be redundant, it helps me slow down and lets me think about what the ideas really mean. Other times I’ll just rewrite things in my own words in a way that makes sense to me.

A semester’s worth of notes!

As for the content itself, my notes usually follow a “definition then theorem then proof” outline, simply because that’s how material is often presented in the lecture. But sometimes it’s hard to see the forest for the trees (i.e. it’s easy to get lost in the details), so I’ll occasionally write “PAUSE!” or “KEY IDEA!” in the middle of the page. I’ll then take the time to write a mini exposition that summarizes the main idea of the previous pages. I’ve found this to be especially helpful when looking back at my notes after several months (or years) have gone by. I may not have time to read all the details/calculations, so it’s nice to glance at a summary for a quick refresher.

And every now and then, I’ll rewrite my rewritten notes in the form of a SaiBlog post! Many of my earlier posts here at Math3ma were “aha” moments that are now engrained in my brain because I took the time to SaiBlog about them.

STEP 4

Do homework problems

Once upon a time, I used to think the following:

How can I do problems if I haven’t spent a bajillion hours learning the theory first?

But now I believe there’s something to be said for the converse: 

How can I understand the theory if I haven’t done a bajillion examples first?

In other words, taking good notes and understanding theory is one thing, but putting that theory into practice is a completely different beast. As a wise person once said, “The only way to learn math is to DO math.” So although I’ve listed “do homework problems” as the last step, I think it’s really first in terms of priority.

Typically, then, I’ll make a short to-do list (which includes homework assignments along with other study-related duties) each morning. And I’ll give myself a time limit for each task. For example, something like “geometry HW, 3 hours” might appear on my list, whereas “do geometry today” will not. Setting a time gives me a goal to reach for which helps me stay focused. And I may be tricking my brain here, but a specific, three-hour assignment sounds much less daunting than an unspecified, all-day task. (Of course, my lists always contain multiple items that take several hours each, but as the old adage goes, “How do you eat an elephant? One bite at a time.”)

By the way, in my first two years of grad school I often worked with my classmates on homework problems. I didn’t do this in college, but in grad school I’ve found it tricky to digest all the material alone – there’s just too much of it! So typically I’d first attempt exercises on my own, then meet up with a classmate or two to discuss our ideas and solutions and perhaps attend office hours with any questions.

As far as storage goes, I have a huge binder that contains all of my rewritten notes*** from my first and second year classes. (I use sheet protectors to keep them organized according to subject.) On the other hard, I use a paper tray like this one to store my lecture notes while the semester is in progress. Once classes are over, I’ll scan and save them to an external hard drive. I’ve also scanned and saved all my homework assignments.

Well, I think that’s about it! As I mentioned earlier, these steps were only my ideal plan. I often couldn’t apply them to every class — there’s just not enough time! — so I’d only do it for my more difficult courses. And even then, there might not be enough time for steps 1 and 3, and I’d have to start working on homework right after a lecture.

But as my advisor recently told me,”It’s okay to not know everything.” Indeed, I think the main thing is to just do something. Anything. As much as you can. And as time goes on, you realize you really are learning something, even if it doesn’t feel like it at the time.

Alright, friends, I think that’s all I have to share. I hope it was somewhat informative. If you have any questions, don’t hesitate to leave it in a comment below!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

Mathematicians are stunned by the discovery that prime numbers are pickier than previously thought. The find suggests number theorists need to be a little more careful when exploring the vast infinity of primes.

Primes, the numbers divisible only by themselves and 1, are the building blocks from which the rest of the number line is constructed, as all other numbers are created by multiplying primes together. That makes deciphering their mysteries key to understanding the fundamentals of arithmetic.

Although whether a number is prime or not is pre-determined, mathematicians don’t have a way to predict which numbers are prime, and so tend to treat them as if they occur randomly. Now Kannan Soundararajan and Robert Lemke Oliver of Stanford University in California have discovered that isn’t quite right.

“It was very weird,” says Soundararajan. “It’s like some painting you are very familiar with, and then suddenly you realise there is a figure in the painting you’ve never seen before.”

Surprising order

So just what has got mathematicians spooked? Apart from 2 and 5, all prime numbers end in 1, 3, 7 or 9 – they have to, else they would be divisible by 2 or 5 – and each of the four endings is equally likely. But while searching through the primes, the pair noticed that primes ending in 1 were less likely to be followed by another prime ending in 1. That shouldn’t happen if the primes were truly random – consecutive primes shouldn’t care about their neighbour’s digits.

“In ignorance, we thought things would be roughly equal,” says Andrew Granville of the University of Montreal, Canada. “One certainly believed that in a question like this we had a very strong understanding of what was going on.”

The pair found that in the first hundred million primes, a prime ending in 1 is followed by another ending in 1 just 18.5 per cent of the time. If the primes were distributed randomly, you’d expect to see two 1s next to each other 25 per cent of the time. Primes ending in 3 and 7 take up the slack, each following a 1 in 30 per cent of primes, while a 9 follows a 1 in around 22 per cent of occurrences.

Similar patterns showed up for the other combinations of endings, all deviating from the expected random values. The pair also found them in other bases, where numbers are counted in units other than 10s. That means the patterns aren’t a result of our base-10 numbering system, but something inherent to the primes themselves. The patterns become more in line with randomness as you count higher – the pair have checked up to a few trillion – but still persists.

“I was very surprised,” says James Maynard of the University of Oxford, UK, who on hearing of the work immediately performed his own calculations to check the pattern was there. “I somehow needed to see it for myself to really believe it.”

Stretching to infinity

Thankfully, Soundararajan and Lemke Oliver think they have an explanation. Much of the modern research into primes is underpinned G H Hardy and John Littlewood, two mathematicians who worked together at the University of Cambridge in the early 20^th century. They came up with a way to estimate how often pairs, triples and larger grouping of primes will appear, known as the k-tuple conjecture.

Just as Einstein’s theory of relativity is an advance on Newton’s theory of gravity, the Hardy-Littlewood conjecture is essentially a more complicated version of the assumption that primes are random – and this latest find demonstrates how the two assumptions differ. “Mathematicians go around assuming primes are random, and 99 per cent of the time this is correct, but you need to remember the 1 per cent of the time it isn’t,” says Maynard.

The pair used Hardy and Littlewood’s work to show that the groupings given by the conjecture are responsible for introducing this last-digit pattern, as they place restrictions on where the last digit of each prime can fall. What’s more, as the primes stretch to infinity, they do eventually shake off the pattern and give the random distribution mathematicians are used to expecting.

“Our initial thought was if there was an explanation to be found, we have to find it using the k-tuple conjecture,” says Soundararajan. “We felt that we would be able to understand it, but it was a real puzzle to figure out.”

The k-tuple conjecture is yet to be proven, but mathematicians strongly suspect it is correct because it is so useful in predicting the behaviour of the primes. “It is the most accurate conjecture we have, it passes every single test with flying colours,” says Maynard. “If anything I view this result as even more confirmation of the k-tuple conjecture.”

Although the new result won’t have any immediate applications to long-standing problems about primes like the twin-prime conjecture or the Riemann hypothesis, it has given the field a bit of a shake-up. “It gives us more of an understanding, every little bit helps,” says Granville. “If what you take for granted is wrong, that makes you rethink some other things you know.”

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jacob Aron*

Yesterday I received a disheartening 44/50 on a homework assignment. Okay okay, I know. 88% isn’t bad, but I had turned in my solutions with so much confidence that admittedly, my heart dropped a little (okay, a lot!) when I received the grade. But I quickly had to remind myself, Hey! Grades don’t matter.

The six points were deducted from two problems. (Okay, fine. It was three. But in the third I simply made an air-brained mistake.) In the first, apparently my answer wasn’t explicit enough. How stingy! I thought. Doesn’t our professor know that this is a standard example from the book? I could solve it in my sleep! But after the prof went over his solution in class, I realized that in all my smugness I never actually understood the nuances of the problem. Oops. You bet I’ll be reviewing his solution again. Lesson learned.

In the second, I had written down my solution in the days before and had checked with a classmate and (yes) the internet to see if I was correct. Unfortunately, the odds were against me two-to-one as both sources agreed with each other but not with me. But I just couldn’t see how I could possibly be wrong! Confident that my errors were truths, I submitted my solution anyway, hoping there would be no consequences. But alas, points were taken off.

Honestly though, is a lower grade such a bad thing? I think not. In both cases, I learned exactly where my understanding of the material went awry. And that’s great! It means that my comprehension of the math is clearer now than it was before (and that the chances of passing my third qualifying exam have just increased. Woo!) And that’s precisely why I’m (still, heh…) in school.

So yes, contrary to what the comic above says, grades do exist in grad school, but – and this is what I think the comic is hinting at – they don’t matter. Your thesis committee members aren’t going to say, “Look, your defense was great, but we can’t grant you your PhD. Remember that one homework/midterm/final grade from three years ago?” (They may not use the word “great” either, but that’s another matter.) Of course, we students should still work hard and put in maximum effort! But the emphasis should not be on how well we perform, but rather how much we learn. Focus on the latter and the former will take care of itself. This is true in both graduate school and college, but the lack of emphasis on grades in grad school really brings it home. And personally, I’m very grateful for it because my brain is freed up to focus on other things like, I don’t know, learning math!

So to all my future imperfect homework scores out there: bring it on.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

In sum, the sufficient condition (a.k.a. the “if” direction) allows you to get what you want. That is, if you assume the sufficient condition, you’ll obtain your desired conclusion. It’s enough. It’s sufficient.

On the other hand, the necessary condition (a.k.a. the “only if” direction) is the one you must assume in order to get what you want. In other words, if you don’t have the necessary condition then you can’t reach your desired conclusion. It is necessary.

Here’s a little graphic which summarizes this:

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

Here’s one thing to impress your friends with the next time you order a takeaway: new and exotic ways to slice a pizza.

Most of us divide a pizza using straight cuts that all meet in the middle. But what if the centre of the pizza has a topping that some people would rather avoid, while others desperately want crust for dipping?

Mathematicians had previously come up with a recipe for slicing – formally known as a monohedral disc tiling – that gives you 12 identically shaped pieces, six of which form a star extending out from the centre, while the other six divide up the crusty remainder. You start by cutting curved three-sided slices across the pizza, then dividing these slices in two to get the inside and outside groups, as shown below.

Now Joel Haddley and Stephen Worsley of the University of Liverpool, UK, have generalised the technique to create even more ways to slice. The pair have proved you can create similar tilings from curved pieces with any odd number of sides – known as 5-gons, 7-gons and so on (shaded below) – then dividing them in two as before. “Mathematically there is no limit whatsoever,” says Haddley, though you might find it impractical to carry out the scheme beyond 9-gon pieces.

Haddley and Worsley went one further by cutting wedges in the corners of their shapes, creating bizarre, spikey pieces that still form a circle (the image below shows this happening with 5-gons). “It’s really surprising,” says Haddley.

As with many mathematical results, its usefulness isn’t immediately obvious. The same is true of another pizza theorem, which looks at what happens when a pizza is haphazardly cut off-centre.

“I’ve no idea whether there are any applications at all to our work outside of pizza-cutting,” says Haddley, who has actually tried slicing a pizza in this way for real (see below). But the results are “interesting mathematically, and you can produce some nice pictures”.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jacob Aron*

A maths problem previously tackled with the help of a computer, which produced a proof the size of Wikipedia, has now been cut down to size by a human. Although it is unlikely to have practical applications, the result highlights the differences between two modern approaches to mathematics: crowdsourcing and computers.

Terence Tao of the University of California, Los Angeles, has published a proof of the Erdős discrepancy problem, a puzzle about the properties of an infinite, random sequence of +1s and -1s. In the 1930s, Hungarian mathematician Paul Erdős wondered whether such a sequence would always contain patterns and structure within the randomness.

One way to measure this is by calculating a value known as the discrepancy. This involves adding up all the +1s and -1s within every possible sub-sequence. You might think the pluses and minuses would cancel out to make zero, but Erdős said that as your sub-sequences got longer, this sum would have to go up, revealing an unavoidable structure. In fact, he said the discrepancy would be infinite, meaning you would have to add forever, so mathematicians started by looking at smaller cases in the hopes of finding clues to attack the problem in a different way.

Last year, Alexei Lisitsa and Boris Konev of the University of Liverpool, UK used a computer to prove that the discrepancy will always be larger than two. The resulting proof was a 13 gigabyte file – around the size of the entire text of Wikipedia – that no human could ever hope to check.

Helping hands

Tao has used more traditional mathematics to prove that Erdős was right, and the discrepancy is infinite no matter the sequence you choose. He did it by combining recent results in number theory with some earlier, crowdsourced work.

In 2010, a group of mathematicians, including Tao, decided to work on the problem as the fifth Polymath project, an initiative that allows professionals and amateurs alike to contribute ideas through SaiBlogs and wikis as part of mathematical super-brain. They made some progress, but ultimately had to give up.

“We had figured out an interesting reduction of the Erdős discrepancy problem to a seemingly simpler problem involving a special type of sequence called a completely multiplicative function,” says Tao.

Then, in January this year, a new development in the study of these functions made Tao look again at the Erdős discrepancy problem, after a commenter on his SaiBlog pointed out a possible link to the Polymath project and another problem called the Elliot conjecture.

Not just conjecture

“At first I thought the similarity was only superficial, but after thinking about it more carefully, and revisiting some of the previous partial results from Polymath5, I realised there was a link: if one could prove the Elliott conjecture completely, then one could also resolve the Erdős discrepancy problem,” says Tao.

“I have always felt that that project, despite not solving the problem, was a distinct success,” writes University of Cambridge mathematician Tim Gowers, who started the Polymath project and hopes that others will be encouraged to participate in future. “We now know that Polymath5 has accelerated the solution of a famous open problem.”

Lisitsa praises Tao for doing what his algorithm couldn’t. “It is a typical example of high-class human mathematics,” he says. But mathematicians are increasingly turning to machines for help, a trend that seems likely to continue. “Computers are not needed for this problem to be solved, but I believe they may be useful in other problems,” Lisitsa says.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jacob Aron*

Felker prefaces the quote by saying,

Giving up on math means you don’t believe that careful study can change the way you think.

He further notes that writing, like math, “is also not something that anyone is ‘good’ at without a lot of practice, but it would be completely unacceptable to think that your composition skills could not improve.”

Friends, this is so true! Being ‘good’ at math boils down to hard work and perseverance, not whether or not you have the ‘math gene.’ “But,” you might protest, “I’m so much slower than my classmates are!” or “My educational background isn’t as solid as other students’!” or “I got a late start in mathematics!”* That’s okay! A strong work ethic and a love and enthusiasm for learning math can shore up all deficiencies you might think you have. Now don’t get me wrong. I’m not claiming it’ll be a walk in the park. To be honest, some days it feels like a walk through an unfamiliar alley at nighttime during a thunderstorm with no umbrella. But, you see, that’s okay too. It may take some time and the road may be occasionally bumpy, but it can be done!

This brings me to another point that Felker makes: If you enjoy math but find it to be a struggle, do not be discouraged! The field of math is HUGE and its subfields come in many different flavors. So for instance, if you want to be a math major but find your calculus classes to be a challenge, do not give up! This is not an indication that you’ll do poorly in more advanced math courses. In fact, upper level math classes have a completely (I repeat, completely!) different flavor than calculus. Likewise, in graduate school you may struggle with one course, say algebraic topology, but find another, such as logic, to be a breeze. Case in point: I loathed real analysis as an undergraduate** and always thought it was pretty masochistic. But real analysis in graduate school was nothing like undergraduate real analysis (which was more like advanced calculus), and now – dare I say it? – I sort of enjoy the subject. (Gasp!)

All this to say that although Felker’s article is aimed at folks who may be afraid to take college-level math, I think it applies to math majors and graduate students too. I highly recommend you read it if you ever need a good ‘pick-me-up.’ And on those days when you feel like the math struggle is harder than usual, just remember:

Even the most accomplished mathematicians had to learn HOW to learn this stuff!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

Example 5

A sequence of functions {fn:R→R}{fn:R→R} which converges to 0 pointwise but does not converge to 0 in L1L1.

This works because: The sequence tends to 0 pointwise since for a fixed x∈Rx∈R, you can always find N∈NN∈N so that fn(x)=0fn(x)=0 for all nn bigger than NN. (Just choose N>xN>x!)

The details: Let x∈Rx∈R and fix ϵ>0ϵ>0 and choose N∈NN∈N so that N>xN>x. Then whenever n>Nn>N, we have |fn(x)−0|=0<ϵ|fn(x)−0|=0<ϵ.

Of course, fn↛0fn↛0 in L1L1 since∫R|fn|=∫(n,n+1)fn=1⋅λ((n,n+1))=1.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

Suffixes are important!

Did you know that the words

“maximal” and “maximum” generally do NOT mean the same thing

in mathematics? It wasn’t until I had to think about Zorn’s Lemma in the context of maximal ideals that I actually thought about this, but more on that in a moment. Let’s start by comparing the definitions:

Do you see the difference? An element is a maximum if it is larger than every single element in the set, whereas an element is maximal if it is not smaller than any other element in the set (where “smaller” is determined by the partial order ≤≤). Yes, it’s true that the* maximum also satisfies this property, i.e. every maximum element is also maximal. But the converse is not true: if an element is maximal, it may not be the maximum! Why? The key is that these definitions are made on a partially ordered set. Basically, partially ordered just means it makes sense to use the words “bigger” or “smaller” – we have a way to compare elements. In a totally ordered set ALL elements are comparable with each other. But in a partially ordered set SOME, but not necessarily all, elements can be compared. This means it’s possible to have an element that is maximal yet fails to be the maximum because it cannot be compared with some elements. It’s not too hard to see that when a set is totally ordered, “maximal = maximum.”**

How about an example? Here’s one I like from this scholarly site which also gives an example of a miminal/minimum element (whose definitions are dual to those above).

Example

Consider the set

where the partial order is set inclusion, ⊆⊆. Then

• {d,o}{d,o} is minimalbecause {d,o}⊉x{d,o}⊉x for every x∈Xx∈
• e. there isn’t a single element in XX that is “smaller” than {d,o}{d,o}
• {g,o,a,d}{g,o,a,d} is maximalbecause {g,o,a,d}⊈x{g,o,a,d}⊈x for every x∈Xx∈X
• e. there isn’t a single element in XX that is “larger” than {g,o,a,d}{g,o,a,d}
• {o,a,f}{o,a,f} is both minimal and maximal because
• {o,a,f}⊉x{o,a,f}⊉x for every x∈Xx∈X
• {o,a,f}⊈x{o,a,f}⊈x for every x∈Xx∈X
• {d,o,g}{d,o,g} is neither minimal nor maximal because
• there is an x∈Xx∈X such that x⊆{d,o,g}x⊆{d,o,g}, namely x={d,o}x={d,o}
• there is an x∈Xx∈X such that {d,o,g}⊆x{d,o,g}⊆x, namely x={g,o,a,d}x={g,o,a,d}
• XX has NEITHER a maximum or a minimum because
• there is no M∈XM∈X such that x⊆Mx⊆M for everyx∈Xx∈X
• there is no m∈Xm∈X such that m⊆xm⊆x for everyx∈Xx∈X

Let’s now relate our discussion above to ring theory. One defines an ideal MM in a ring RR to be a maximal ideal if M≠RM≠R and the only ideal that contains MM is either MM or RR itself, i.e. if I⊴RI⊴R is an  ideal such that M⊆I⊆RM⊆I⊆R, then we must have either I=MI=M or I=RI=R.

Not surprisingly, this coincides with the definition of maximality above. We simply let XX be the set of all proper ideals in the ring RR endowed with the partial order of inclusion ⊆⊆. The only difference is that in this context, because we’re in a ring, we have the second option I=RI=R.

I think a good way to see maximal ideals in action is in the proof of this result:

As a final remark, the notions of “a maximal element” and “an upper bound” come together in Zorn’s Lemma which is needed to prove that every proper ideal in a ring is contained in a maximal ideal. I should mention that an upper bound BB on a partially ordered set (a.k.a. a “poset”) has the same definition as the maximum EXCEPT that BB is not required to be inside the set. More precisely, we define an upper bound on a subset YY of XX to be an element B∈XB∈X such that y≤By≤B for every y∈Yy∈Y.

So here’s the deal with Zorn’s Lemma: It’s not too hard to prove that every finite poset has a maximal element. But what if we don’t know if the given poset is finite? Or what happens if it’s infinite? How can we tell if it has a maximal element? Zorn’s Lemma answers that question:

‍As I mentioned above, it’s this result which is needed to prove that every proper ideal is contained in a maximal ideal***. It actually implies a weaker statement, called Krull’s Theorem (1929), which says that every non-zero ring with unity contains a maximal ideal.

‍Footnotes

*One can easily show that if a set has a maximum it must be unique, hence THE maximum.

** Here’s the proof: Let (X,≤)(X,≤) be a totally ordered set and let m∈Xm∈X be a maximal element. It suffices to show mm is the maximum. Since XX has a total order, either m≤xm≤x or x≤mx≤m for every x∈Xx∈X. If the latter, then mm is the maximum. If the former, then m=xm=x by definition of maximal. In either case, we have x≤mx≤m for all x∈Xx∈X. Hence mm is the maximum.

*** Note this is NOT the same as saying that every maximal ideal contains all the proper ideals in a ring! Remember, maximal ≠≠ maximum!!

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*Credit for article given to Tai-Danae Bradley*