 In this video I want to define the notion of a hazai diagram, it's sometimes called a lattice diagram. A hazai diagram of a partially ordered set, so X with its relationship forms a poset. A hazai diagram is a graph whose vertices are the elements of X, an example of this you can see down here. It's a graph whose elements are the vertices, the vertices of the graph are X, and then we're gonna draw an edge between two vertices, so we draw something like this. Whenever X and Y are related to each other, but then there's no intermediate element that sits between them, right? So that is X and Y are related to each other, but there's no one between them in terms of this relationship. So we call this a hazai diagram. And so because of this, we draw the small elements on the bottom, we draw the big elements on the top, and the idea is as you go up the edges, this would say that the inequalities are going in this direction. So like something like this. So when you go up the diagram, that's how we say that that element's bigger than another. And so let me present you some examples of a hazai diagram, and it's a visualization of the partially ordered set here. So let's take a very simple example. Let's take X to be a three element set. We'll just call the elements A, B, and C for the sake of it. We've seen previously that the power set on X forms a poset with respect to set containment. This set here will contain two to the three, that is eight many elements. There are eight subsets of X, which include the empty set itself, which we might call that the trivial subset. And then there's the improper subset, that is this is just X itself. And then there's gonna be three subsets of order two, and there's three subsets of order one. Now, if we draw the, if we draw the containment's here, right? So if you take the elements, like let's take a singleton for example, the one, the set that contains A. If you have a singleton, it only has two subsets. There's itself, and then there's the empty set. And because of, it just contains one element, there's no intermediate sets between the singleton A and the empty set. And this is, and so we're gonna draw a line between the singleton A and the empty set. We're gonna do the same thing for the singletons B and C. And when one draws a, the posi diagram associated to the power set, singletons are, well, the empty sets at the very bottom. And then you'll see singletons always above that. The next thing is, if we take the element, the subsets of order two, I should say of cardinality two, like take AB for example, it's immediate predecessors are gonna be the singleton A and the singleton B. Notice there is no line between AB, the set, and the singleton C because C is not a subset of AB. So we don't draw anything right there. Similarly, if we take the set AC, it has as its predecessors A and C. And if you take the subset BC, it has as its predecessors B and C, like so. Then that takes care of all the sets of cardinality two. Then as the largest set has cardinality three, the sets of cardinality two are gonna be the immediate predecessors of the set X itself. And so you get this haze diagram, which honestly, if you wanna think about it three-dimensionally, this looks like the vertices of a cube position, maybe from a different perspective, right? I should also mention that when it comes to a haze diagram, as we're trying to graph a partially ordered set, the partially ordered set has the transitivity property. So if you know that the empty set is a subset of A, and A is a subset of AB, that would imply that the empty set is a subset of AB. But we don't need to draw a line from the empty set to AB because we don't wanna get at this graph to be too messy. It can be inferred that because there's a path from the empty sets to AB, that AB is bigger than the empty set. And when I say there's a path, it must always be an upward traveling path. So for example, the empty set is a subset of the whole set because there's an upward path going there. On the other hand though, I can't do something like the following, B goes to AB, AB goes to ABC, and then goes to AC, right? That would not say that AC is a super set of B because all paths must be going upwards. I had a downward trajectory at one point, so that type of path wouldn't matter. That's not what we mean by the transitivity property. So let's look at some other examples. And now this time I'm gonna focus on the poset of subgroups. This is the main reason why we're talking about these Hase diagrams. We wanna look at the partially ordered set of subgroups. So in this regard, we're actually gonna return to section 3.3 in Judson's textbook. Consider the group Z6. By a previous observation, I claimed that the subgroups of Z6 were the following. You of course get Z6 itself, you always get the improper subgroup. You always get the trivial subgroup, which is just a singleton of the identity. But I also mentioned that there were two other subgroups. There is this subgroup of order three that contains zero, two, and four. And then there's this subgroup of order two that contains zero and three. I mentioned how Z5 could not be made into a subgroup of Z6. And in fact, by arguments that one could try to investigate, there is no other subset of Z6 that would form a subgroup. These are the only four. And we can see them in the Hase diagram that these two right here are considered minimal subgroups because the only proper subgroup of these two is the trivial group itself. So these minimal subgroups are the ones we're gonna put on the bottom, these minimal subgroups. You have a minimal subgroup of order three and you have a minimal subgroup of order two. Now there are no other proper subgroups of the group itself. And so we see there's this line connected between the subgroup of order two and the trivial subgroup because there's no groups between them. Same thing here. There's no subgroup between the trivial subgroup and the subgroup of order three. But likewise, there's no subgroup larger than the subgroup order two other than the whole group itself. And the same can be said for the subgroup of order three. Let's give a little bit more interesting picture. Again, this one, we're not gonna necessarily prove all the details right now. I'll leave the verification kind of up to you or this will be a lot easier when we talk about cyclic groups and Lagrange's theorem. But for the time being, let's consider the group Z24. So this would be addition, modulo 24. It's a group of 24 elements. The subgroups are gonna look like the following. There is gonna be a group, the whole group, it has order 24. There's the trivial group which has order one, right? And so I'm gonna list these things as we go. So there's a group of order one. There'll be a group of order two, which contains zero and 12. There'll be a group of order three, which contains zero, eight, and 16. These subgroups of order two and three have no smaller subgroups inside of them other than the trivial group itself. So we draw lines that connect them like this. Next, there is gonna be a subgroup of order four. And there will likewise be a subgroup of order six that you see right there. The subgroup of order four will contain the subgroup of order two, and the subgroup of order six will contain the subgroups of order two and of three. So you see that from the line segments we've drawn right here. Next, there'll be a subgroup of order eight that you see right here. And there will also be a subgroup of order 12 that you see right there. The subgroup of order eight will contain the subgroup of order four, which by transitivity will contain the subgroup of order two and the trivial subgroup. The subgroup of order 12 will contain the subgroup of order four and the subgroup of order six and whichever path you go, this will contain the subgroup of order two, the trivial subgroup, but you also get the subgroup of order three. And then of course the whole group will contain them all, like right here. So there's a subgroup of order 24. Now, one thing you might notice here is if you look at the orders of the subgroups you get one, two, three, four, six, eight, 12, 24. These are exactly the divisors of 24. That it's not a coincidence. That's something we're gonna talk about more in the future. You'll also notice that if you look at, for example, the subgroup that contains eight elements, look at the numbers here, zero, three, six, nine, 12, 15, 18, 21. These are just multiples of three. And notice of course that three times eight is equal to 24. Yep, you saw it there. Look at this one right here. This is a subgroup of order six and it contains all the multiples of four, right? Zero, four, eight, 12, 16, 20. And notice when you take four times six, that's equal to two, two. I'll leave you in suspense right there. There is actually some interesting connection going on here. And we'll study this type of group. What's going on here? That there's a lot of interesting structure that comes from the group itself, which again I'm alluding to at the moment, but we'll do some more details of this in the future. There's some interesting things that you learn about when you start studying groups, there's subgroups and other related questions to that.