 Welcome back to our lecture series Math 4220, Abstract Algebra 1 for students at Southern Utah University. As usual, I'll be your professor today, Dr. Angela Miseldine. This is our first video for lecture 11, which we're going to talk about the idea of a haze diagram. Although there's a little bit of background we're going to have to get into before we get directly to the haze diagrams. So if you've been following us with this series, you might be a little bit surprised at the following heading. Section 19.1 from Tom Judson's textbook, what just happened, weren't we just like in chapter 3? Well, this is the background I'm referring to. Section 19.1 chapter 19, in fact, is about partially ordered sets, lattices, the algebra of partially ordered sets, and the like. So I want to just do a quick review of partially ordered sets before we talk about the idea of a haze diagram because, again, the haze diagram is going to be a lattice which naturally comes from these objects here. So in summary stack, a partially ordered set is kind of like an equivalence relationship with one very important tweak. So first of all, a partial order is a relation on the set X. So we say that a partial order, say less than or equal to here, is a relationship on the set X itself. So what that means is this is kind of funny looking here, but the partial order, the relation less than or equal to is a subset of the set X cross X right here. Now, if you are curious, this isn't a proper less than, this is actually the symbol backslash p, r, e, q, sorry, e, c, e, q, right here. And the idea is the p, e, the p, r, e, c is short for predecessor. If you drop off the equals there, you always get the predecessor without the line under it. But I always put the equal here. So you have predecessor. If you want to go the other way around, you can kind of like this curly thing. It's hard to write with my pen here. If you want to go the other way around, this would backslash s, u, c, c, e, c. So successor, predecessor versus successors there. So a partial order is a relationship on a set X that has size three conditions and these conditions will look very, very similar. So the first one is the reflexive property. Oh, okay, I see that. So that says that X will be related to X for every element inside the set. The third condition, what happened number two, we'll come back to that one. The third condition is the transitivity property, right? So if X is related to Y and Y is related to Z, then X is related to Z. And partial orders here are trying to capture the idea of a symbol like less than or equal to. That's why we're picking something that kind of looks like less than or equal to, but isn't exactly. So that we feel like it could be different, but they're very similar, right? The fact that it's less than or equal to means it should be reflexive. When it comes to partial orders, they should be transitive, right? If X is less than Y and Y is less than Z, then X should be less than Z, right? If it just keeps on getting bigger, that should be retained, right? So I skipped the second one. So let's come to this one. Because if you look at the first one and the second one, this is exactly what we expect for an equivalence relationship. Now for a partial order, we actually tweak the symmetric property. We don't actually take the symmetric property, we take what's called the anti-symmetric property. So if X is related to Y and Y is related to X, then in fact, it must have been that X is equal to Y. So remember the symmetric property for equivalence relationships. This would say something like if X was related to Y, then Y is related to X, right? So the symmetric property says that whenever a relationship holds, the symmetric relationship holds automatically. We're not saying that. What we're saying is if the symmetric relationship holds, it actually must have been the case that the two things are equal. The only way you can reverse the direction is if with equality. And if you think of the usual less than or equal to symbol, if X is less than or equal to Y and Y is less than or equal to X, which is basically saying that X is less than or equal to Y and Y is less than or equal to X, right? The only way that could happen is this would force equality, right? X equals Y. That's the anti-symmetric property. And so if X is equipped with a partial order, we call it a partially ordered set. Although some people call this a poset for short, partially ordered set. Some people actually take offense to the word poset. I think it's too crude or something. I'm okay with either one. Partially ordered set versus a poset, good with either one. So a set equipped with a partially ordered set, we call it, sorry, a set equipped with a partial order. Sometimes it's called a partial ordering. This is a poset or a partially ordered set. So if two elements belong to the set X and there's a relationship between them, X is less than or equal to Y. That's usually how you read it. X is less than or equal to Y or Y is less than or equal to X. One of those things happens. Then we say that the elements are comparable. And if no such relationship exists between X or Y, there's no relationship between X and Y in either direction. Then we would say that the two elements are incomparable or incomparable. However you want to pronounce that word. And so the archetype of what a partial order is, is in fact the partial order that exists on the real numbers. We say that two real numbers A and B satisfy this relationship. We say that A is less than or equal to B. Whenever the difference A minus B is non-negative. So if the bigger number is subtracted by the smaller number, that actually will give you something positive. Of course, if they're equal, that'll give you zero. So we say that B is bigger than A or equal to when their difference is non-negative. Zero is a possibility. Now you can restrict partial orders from one set to a smaller set. This works out pretty easily. We talked about in the previous lecture how you can restrict a binary operation to talk about a subgroup. That's not such a big deal when it comes to a partial order. If you look at a subset, you just retain whatever relationships existed previously. And so we can make the rational numbers into a partial order set. The integers, the natural numbers using the same partial order that existed on the real numbers. Now I should caution you that this partial order does not extend to the complex numbers. We don't have a well-defined notion of when a complex number is less than or equal to another one. At least not by extending this partial order right here. So the real numbers is an example of a poset. Another archetype of what a poset is all about is the power set. So given a set X, its power set, which is commonly referred to as P of X, but honestly in the literature, it's probably getting more mentioned as two to the X, at least in common torques in set theory. That's how we like to look at it, two to the X here. And the reason I say this is from common torques is because the power set, if X has a cardinality of N, the power set is going to have a cardinality of two to the N. And so we oftentimes use two to the X in that situation to develop the power set. It's also borrowing from the notation that if you have the set Y to the X, this is the set of functions from X to Y, like so. And so this two to the X is thinking that a subset can be identified with a function from two to X, whereby two here, I mean, you have the options of zero and one. These binary sequences there. So we don't need to go into the common torques too much, but you'll often see two to the X used to describe the power set of some set X. I do believe in Judson's textbook, he uses P of X, so we'll probably stick with that when it is commonly used. The power set can be made into a partially ordered set, where the partial order is going to be the set containment symbol. Now be aware that set containment is a partial order. It satisfies the conditions. It's reflexive, right? Because what does the subset symbol mean here? So we say that A is a subset of B if and only if for all little a's and a's, it follows that little a is inside of B as well, right? So that's what it means to be a subset. So if you take an element of X, if you take an element X inside of A, it's an A. Therefore, since X was chosen arbitrarily, A is a subset of A, the reflexive property holds. What about anti-symmetry? Suppose that A is less than or equal to B and B is less than or equal to A, right? So A is a subset of B and B is a subset of A. Well, that implies that the two sets are equal to each other. That's actually how we define set equality, that two sets are equal exactly when A is a subset of B, that is every element of A belongs to B and B is a subset of A. That is every element of B belongs to A. So yeah, that one's kind of like automatic in that regard. And then transitivity, if A is a subset of B and B is a subset of C, we would want that A is a subset of C, right? So take an arbitrary element of A because A is a subset of B, every element of A will be an element of B. But since B is a subset of C, every element of B, which includes X, is an element of C. And as X was an arbitrary element of A, this then shows that A is a subset of C. And so therefore the set containment symbol is an example of a partial order. And now unlike the previous example, this actually gives us a reason why do we call it a partial order? Why partially? What's so partial about it? Well, if you were to take a set like say X equals the set 1, 2, 3, then some of the subsets of X would be like Y could equal 1 and Z could equal 2, 3. So these are both subsets of X, right? They both are subsets of X, so they belong to the power set. But how do these compare? It's not true that Y is a subset of Z. That's actually false. It's also not true that Z is a subset of Y. Y doesn't belong to Z. It's not a subset of Z because one's not inside of Z. But Z is not a subset of Y because 2 doesn't belong to Y. And so this would be an example of two elements which are incomparable. And that's why we call it a partial ordering. That we don't say that Y is bigger than Z or Z is bigger than Y. There's no comparison there. And so because of the possibility of in comparisons, we do call these things partial. Now, kind of basically using the same reasoning as we did before. This is the main reason why we're starting talking about it in this section. If you have a group G, we could take the family, the set of all subgroups of that group, and then using that symbol less than or equal to right. Then by basically the same arguments as before, the set of subgroups of a group is a partially ordered set in terms of the set containment symbol, which we draw it a little bit differently for subgroups. So that way to indicate that a subset is a subgroup. It's a different symbol, but it'll also be a partially ordered set. And this is the post set that we're very much interested in right now. And then just as a last example, sort of like another canonical example, if you have a set of positive integers, then and you take the relationship of divisibility, where X could be any positive. So X is going to be the positive divisors of some positive integer N. Then with respect to divisibility symbol, this also forms a partially ordered set. I'm actually going to leave it up to an exercise to the viewer here to prove that one. I don't want to steal the thunder there, but these give some important examples of partially ordered sets.