 Welcome back to our lecture series Math 4230, Abstract Algebra 2 for students at Southern Utah University. As usual, I'll be a professor today, Dr. Angea Missildine. Now in lecture 38, we want to continue to develop notions we have about lattices, ultimately building up to the most structured lattice possible, that is of a Boolean algebra. We're not going to define a Boolean algebra in this particular video, but we will do that by the end of lecture 38. I want us to remember what was the original goal of introducing lattice theory in the first place. We basically had two examples in mind. As we're developing lattice theory, we wanted to maximize the idea of set algebra. Set algebra here being that we have some set, some set called x, and then we look at the algebraic structure where we take the power set of x and equip that with unions and intersections. What type of algebraic structure is that? We mentioned how the power set with union intersection does form a lattice. But it turns out which of course a lattice means we have two associative commutative operations for which every element is ident potent with respect to both operations and the operations absorb each other. We also mentioned of course that with a lattice, the axiom of ident potency is actually redundant. You can prove that from associativity commutativity and absorption. But check out that other video if you want to see the details of that. Lattices are trying to generalize the idea of set algebra, this structure right here. But there's also a very important alternative that we're also trying to generalize. This actually suggests the name Boolean algebra, which we'll define at the end of this lecture right here. We want to actually maximize the notion of Boolean logic. Boole was one of the first people who tried to take the study of logic and study it algebraically. With Boolean logic, you do have a lattice structure. For what you have is you actually only have two elements inside your lattice there. You have true and false. Those are the two possible evaluations you can attach to a statement. A statement's either true or it's false. You have two operations. You have the operation of or and the operation of and. This forms a lattice structure also. Now typically these symbols or and and, we can write the words or or and, but we also sometimes use the symbol this for or and this for and. For which if you notice, those are exactly the two symbols we use for join and meet inside of our lattice theory. Because after all, we're trying to generalize this idea of Boolean logic in addition to set theory. These are both examples of lattices, but it turns out there's more structure there. These two objects set algebra for some power set and Boolean algebra, excuse me, Boolean logic. These are both examples of Boolean algebras. There's more structure just beyond what the lattice structure is. And so what are those missing pieces? We want to try to gather that in this lecture right here, lecture 38. So in this video, we want to talk about the idea of identities and particularly what does it mean for a lattice to be bounded? Now, before we do that, let me offer you an example of a semi lattice. So remember a semi lattice we're only considering one of the binary operations. And so for the sake of discussion, let us consider just the join, just the join operation there. So we have some set over here. We'll call this set S for semi lattice and we have the join operation here. So let's take some elements and I'll just call them by alphabetic letters, A, B, C, D. I'm gonna skip E for a moment for reasons that'll become clear in just a moment. I'm gonna come and call this element at the top theta. Again, for reasons that'll make more sense a little bit later on. But we have the semi lattice. I drew the semi lattice. So I actually drew it as a Hase diagram for which a Hase diagram gives us a semi lattice if and only if there is a least upper bound for each element. So when you look at like A and B, they do have a least upper bound. It's this element D right here. If you look at A and C, it's least upper bound is C because C is bigger than A. The least upper bound of A and D is D. The least upper bound of A and F actually would be theta in this situation. And then likewise, the least upper bound between A and theta is theta itself. So that gives you one example. You can do all the rest to verify that the semi lattice diagram does in fact give us a semi lattice structure, this S join structure right here. Now this set, I'm gonna move it up a little bit. This set doesn't have an identity element. That is, there is no element, there is no element we'll call it E, which is why I skipped it earlier. There's no element E inside of S such that for all X inside of S, we have that E join X is equal to X join E which is then equal to X. That's not true for a universal element. Now I want you to be aware here that if you take for example, the letter A, right? There is an element that kind of acts like an identity because notice if I take A join A, that does give you back A, right? And similarly, if you take like A join C, that gives you C, A join D gives you D, there's an A join theta gives you theta. So A kind of acts like an identity, at least locally speaking, but there are some issues of course. If you take A join B, that's actually equal to D, which is not B, so A is not the identity for B. Same thing happens with F here. If you take A join F, that's gonna give you theta, which is not F. So A is not an identity for every element and that's what an identity has to be. An identity is a global element, not a local element. So A cannot be the identity for everything. B is also a contender, but the same thing happens. If you take A join B, you don't get back. A, you get back D, something else. And no other element can act like a universal identity. So this similatus, as we draw on it, has no identity element. Now the nice thing about similatus is it's very easy to define an identity. That is, you just throw in an extra point, connect it to all the bottom elements. That is the things at the very bottom of your lattice. You can join in this element, we call it E. And now with this structure now in mind, E is in fact the identity for the whole similatus. So if you take E join X, you always get back X. It doesn't matter what you choose. Because E by our construction is smaller than everything, you put on a minimal element to the similatus, that then becomes the identity with respect to the join operation. And so that's something I wanna mention here, that the join identity, if you have one, the join identity is equal to the minimum element of the similatus. Now by duality, if you wanted to do the same thing with a meat similatus, you'll have a meat join if you have a maximum element, right? So your meat, what did I say? Did I say meat join? The meat identity, that's what I should have said, the meat identity is gonna coincide with the maximum element. So that actually comes to this element theta we have before. Now without the E, we actually didn't have a lattice structure because A and B didn't have a greatest lower bound. They didn't have a lower bound at all, let alone a greatest one as well. So the inclusion of the identity for the join actually forms a lattice now, so that now when you look at this picture, this lattice, every pair has now a greatest lower bound. Because at the very least, you have the identity which is a lower bound for everything. This identity E is less than or equal to X for all elements X there, so it's a minimum element. Every element, since you have a join identity, then has a lower bound for which then we worry about is there a greatest lower bound? In fact, there is one here. Now conversely, this element theta, which is at the top of my lattice, it's gonna be greater than every element X and it's going to be the meat identity. Because if we look at something like you take any element X and you meet it with theta, since theta is bigger than X, this is going to equal X in that situation. And so the presence of identities inside of a lattice, whether you have a join identity or a meat identity, the presence of those has to do with the partial order. The join identity will be the minimal element of the lattice and the meat identity will be the maximum element of their lattice. And therefore, we say that a lattice is bounded if you have identities for both of your operations. So we say that a lattice is bounded above if we have a join, excuse me, a lattice is gonna be bounded below if it has a join identity because that gives you a minimal element. And so that would be something like this. Let's call that element zero here. So X join zero is equal to X, which is equal to zero join X. So this zero element is the join identity. And so you're gonna, in this case, you're gonna be bounded below, bounded below because that join identity is the minimum element of that lattice. And then conversely, we have some second element, we'll call it one, which is going to be the meat identity so that when you take X, meat one, that equals X, which equals one, meat X. Now, admittedly, since lattices are commutative, you don't necessarily need these second statements here, but to make these actions independent of each other, we always require the identity to be two-sided. That way, if we wanna generalize to some type of non-commutative lattice, what have you. We aren't gonna do that in this lecture series, but it's just good practice to do so. Now, as we observed earlier, the meat identity is gonna be the maximum element of a lattice. So if you have a meat identity, that means your lattice is bounded above. If you have a join identity, your lattice is gonna be bounded below. If your set is bounded above and bounded below, we call it a bounded set. And therefore, a bounded lattice is a lattice with both a lower bound and an upper bound. And those lower bounds will have to be identities for the two operations since the operations are so closely related to the notions of order for this set. Now, why did we call them zero and one? Now, with my original semi-lattice, I called this element theta because theta kinda looks like a zero, but it's not quite a zero. So I didn't wanna tip my hat too quickly there, tip my hand, I should say. And why did I call this element E? Well, I called that element E because in group theory, it was very common for us to call the identity of the group E in that situation, you know, coming from the German tradition. So I was trying to be sort of foreshadowing that. Theta and E are symbols we often use for identities. And those are exactly the identities of the lattice, the maximum element and the small semi-lattice there. But why are we calling them zero and one? Well, it's actually a very common tradition that with lattice theory, since it kind of is an analog to ring theory, instead of using the symbols meet and join, they oftentimes use addition and multiplication for which join is sometimes denoted as addition. That is, instead of using the join symbol, you actually use a plus sign. And similarly, in lattice theory, meets are often identified using multiplication. This is particularly true when you look at bullied algeas, very, very common tradition. So that, okay, the meet actually can be written as multiplication or typically juxtaposition is how you denote meet in that situation. Now, that's very common practice. It's not universal practice. Many people do distinguished lattice operations from like a ring operation as we use strictly different symbols in that situation. But expect, particularly for a student's perspective that this lecture is at the very end of abstract algebra two for which we spent most of the semester, most of this lecture series studying rings, I would hate for us to get too confused. So we are gonna use different symbols in that situation. But because of this tradition, if you think of join actually with a plus sign, then the additive identity of addition is zero. And so it then is common tradition to denote the identity of the join operation as a zero element. Even if you write joins with join symbols and not with addition, it's still common tradition to write it using a zero there. And likewise, if you write meet instead of multiplication, then the multiplicative identity is one. And so by that tradition, one is the common symbol to represent the meet identity. So the meet identity is one, the join identity is zero. And that's gonna be common tradition. And we're gonna do that as well in this. And we'll do that in our discussion of lattices as well. So let us prove some statements about identities inside of a bounded lattice. Now, one thing I'm gonna comment without proof is uniqueness, right? Identities inside of identities are gonna be unique here. And this follows from the fact that we have a two-sided identity. The argument doesn't change. We did this back in group theory for math 42, 20 abstract algebra one. If you have two identities, right? So if you have some identities such that zero joined x is equal to x and you have some second identity. And so it's a two-sided identity x join zero right here. And if you have a second identity called theta, theta join x is equal to x, which is equal to x join theta, right? If you have a two-sided identity, then the thing is to consider what happens when you take the identities on each other. Well, since zero is an identity, when it acts on theta, you're gonna get back theta. But since theta is an identity and you act on zero, you're gonna get back zero. So zero and theta are one the same thing. You are using two-sided identities because you're using that zero as a left identity and theta is a right identity. Like so, I guess I said I wasn't gonna do it, but then whoopsie daisy, I just did it. So that's a result that's gonna be true for binary operations in general. If you have an identity, a two-sided identity, that identity has to be unique, all right? Because basically what we have here is every left identity has to equal the right identities if they're present. So we have uniqueness of identities. That's very nice. Same proof that we used in group theory here. Now, I actually wanna prove this, I wanna prove, I mean, I've stated it intuitively. Let's actually prove this statement about ordering. Let's prove that the join identity, let's prove that the, no, I'll write that out. Let's prove that the join identity, which of course is equal to zero, let's prove that this is equal to the maximum element of every lattice. And let's likewise prove that one which is the meat identity. Oh, I wrote that backwards, didn't I? Fix that one, sorry. The join identity is the minimal element. So it's at the bottom of the lattice. And then the meat identity, this element is at the top of the lattice. This is a maximum element in that situation. So zero is the minimum element, one is the maximum element. And so we wanna prove that. Now, to do that specifically, we wanna show that x join one is always equal to one and x meet zero is always equal to zero. So based upon how our partial order is defined, if x join one is equal to one, that means that x is less than or equal to one. So one would be a maximum element there. And if x meet zero is equal to zero, that means that x is greater than or equal to zero. That's true for every x. So that gives us the max of a minimum conditions. So we just need to prove these identities algebraically and these order theoretic consequences will then be immediate from them. All right, so how are we gonna do that? So note, if we take one join x, okay, that's the same thing as one join one meet x. And that's because this last one right here, one is the meat identity. So x, I can substitute x with one meet x like so. Then when you look at this statement right here, one join one meet x, the absorption axiom comes into play here. And so this simplifies just to be one in that situation. So we then get that one is in fact this maximum element. That gives you the first statement right here by commutivity, the other statement applies as well. And I'm not gonna prove the second one because the second one actually follows from duality. The dual argument where you interchange one with zero and you interchange all the meats with joins and the joins with meats will give you the exact same proof so the principle of duality works here now as well. So one and zero are now interchangeable when you make dual arguments. Now, let me mention a few examples here. Not all lattices are bounded. There do exist lattices that don't have identities. Now, I gave you a first an example of a semi lattice that didn't have an identity, but that wasn't a lattice. It turns out by adding the identity actually made it into a lattice. It is true that all finite lattices are bounded and the argument is basically the following. You're gonna take an element, we're gonna take as our element say one, we're gonna take the join of all elements inside of our lattice and we can define zero similarly. You can take the meat of every element in the lattice and that's gonna equal zero one in that situation. Well, why is that? Well, you're gonna take one in that situation. It's gonna be the maximal element, right? This has to be the maximum element because if I take X and I join it with everything inside of that, let's not use X that might give us the wrong impression there. Let's use some elements say Y. If you take Y join the join of everything in the group, in particular what's inside the lattice, excuse me. At some point there is gonna be an L, excuse me, there's a Y inside of the lattice there. So I can rewrite this thing as Y join, Y join everything else. And as your idempotent, I don't even have to remove it, Y from consideration. It's already in there. And so I'm basically utilizing the fact that big VX is equal to Y VX here because by the idempotency commutativity, associativity, it's all the same thing. So we can absorb the Y because of the idempotency condition. It's all in though, like when we look at this thing, this is gonna look like some X1, some X2, some X3 all the way through to some XN. We have everything in there. One of those elements is equal to Y. So by associativity, by commutativity, I can bring it up. You're gonna get at some point a Y join Y which is just equal to Y, you can put it back in. If you take the join of everything, when you join that anything else, since you already have it and since your idempotent, you don't change it. And so that's your maximum element. This right here is our maximum element. And so by the previous condition, the maximum elements are one and the same thing with these identities. So every finite lattice, because you can, since it's a finite lattice, it has to have a max, it has to have a min. If you had two different maxes, you could take their join together and that would give you a new max. If you had two different minns, you could take their meat and that would give you a new min. So a bounded lattice always has identities, but if, excuse me, a finite lattice always is bounded and hence always has identities. But you can have infinite lattices which that's not the case. Like for example, if you take the integers where your operations is the max and min, that's gonna give you a picture that basically looks like the following. You're gonna get this infinite chain of elements where you can think of like this is zero, this is one, this is two, this is negative one, like so. Max makes you go up, min makes you go down. So if we take the max of zero and one, we're gonna get back a one. If we take the min of zero and one, that's gonna give us back a zero. Now be aware, I'm using zero and one to represent the integer zero and one. I'm not representing the identities of this lattice. It is a lattice, but it doesn't have identities not built into it. So such an example can happen because it's infinite. Now, like we did with the Simi lattice at the very beginning of this video, we can always add in identities if we want to. Like we could throw in the identity for the minimum would be infinity and we could throw in the identity for the max operation which would be negative infinity. As we can always glue on top a maximum element, like a definite maximum element and a definite minimum element and those would be the identities. So every, every lattice can be extended into a bounded lattice by adjoining identities if they're not already there. And this really doesn't change the lattice that much. And so for the oftentimes we can assume lattices are bounded even when they're not because of this type of construction. Now the lattice that we were most interested in is the power set lattice with the unions and intersections. This lattice of course is always bounded. They always have identities there for which the zero element coincides with the empty set and the one element coincides with the whole set itself. All right. So the example we really cared about it has, it always has identities in that situation. And so notice and one more example here and we'll finish up this video here. We've mentioned this idea of bounded above and bounded below. You don't always have to have both lattice, both identities but you can come up with examples which have only one and not the other. So take for example, the positive integers. So, and we're gonna organize them this time by not min and max but by least common multiples and greatest common divisors. So in that situation, our picture would look something like the following. You have one on the bottom. Then you're gonna have basically every prime number. So you have like two, you have three, you have five, you have seven, et cetera. You have all the prime numbers. These are gonna be elements that are sitting above one in our partial order. Then we have other elements. We'll have things like four, which is two squared. We'll have, you know, nine, which is three squared. We'll also have things like two times three which is six. We'll have things like 10, which is two times five. And this thing starts to get messy very quickly. You'll have like a 25 in there, which is five squared. We would have a 15, which is three times five. So again, this picture gets messy very, very quickly. It's an infinite lattice. It's very difficult to draw it. But this infinite lattice, what type of identities do we have? We have a join identity. That is, our join operation is the least common multiple. There is an element. When you take the least common multiple of one with any other element, one is that element. The least common multiple of one and x is always equal to x, that number. And so we do have a join identity, which is one, which is kind of funny, because I said earlier that the join identity we always talk about as zero. So in this case, we have zero equals one. But we also have that there's no, there's, I mean, of course, I mean, that's a joke there. One is the join identity. It's the LCM identity for this structure. But there is no upper bound, right? There is no maximal element. There is no meet identity. So that if you take the GCD of two elements, there's no fixed element E, such that when you take it with x, you always get back x. You could always do something else. You don't have this maximum element in that situation. Now, if you wanted to, you could add a maximum element. You could. And in fact, if you look at the natural numbers with regard to LCM and GCDs, there actually is a maximum element in that situation in zero, believe it or not. So this is a really weird lattice because one is zero and zero is one in that situation. Cause remember zero is the join identity, which is actually the integer one. And one is the meet identity, which is in this case, the natural number zero. But you can do that. If you take the GCD of zero and x, this always gives you back x, the way that GCDs are defined. Cause honestly, zero is divisible by everything. In particular, it's divisible by x. The largest divisor of x is x itself. So you can always join in an identity if you want to. So be aware that if you take just the pods of integers with LCM and GCD, you are only bounded below. You can add in the maximum element to get the meet identity, like we did with the natural numbers. And by duality, if you take the dual lattice to what we just have here a moment ago, you can then give us an example of a lattice, which is bounded above, but not bounded below. But even if you're not bounded below, you can always add in that lower element and then give you an identity in that situation.