 In the previous video we introduced the notion of a semi-lattice, which I claimed was basically half of a lattice. So it seems appropriate that we should actually define what a lattice is. So if a semi-lattice is to a group, then a lattice is to a ring. That is, a semi-lattice has one binary operation that satisfies certain conditions, certain axioms. A semi-lattice will have two binary operations for which they then satisfy certain axioms, which we then will define right now. So a lattice will be a set L equipped with two operations, for which the first operation we call it the join operation, and then the second one we call it the meet. Based upon the previous lecture with semi-lattices, how there's this natural correspondence between a semi-lattice and a partially ordered set, we can instead actually refer to this join instead as the least upper bound operator, in which case then the meet then becomes the greatest lower bound operator, if you want to think in terms of the partial order. These symbols do kind of look like unions and intersections, and we're trying to generalize that with regard to set operations. So a semi-lattice is this set with two operations, meet and join, that satisfy the following quote unquote four axioms. I say they're four axioms, but each axiom actually comes in pairs, so one might actually argue there's eight axioms, but still we put them into groups of two, so we're going to say four axioms. So the four axioms go in the following way. We require that both operations, meet and join, are associative operations. So for any elements x, y and z, you have that x join y join z is the same thing as x join y join z, and then you do the same thing for the meets. That x meet y meet z is equal to x meet y meet z. So these are two associative operations. We require that also the two operations are commutative. So for any elements x, y inside the lattice, we require that x join y is equal to y join x, and we require that x join y is equal to, let's not join, excuse me, x meet y is equal to y meet x, right? So both of these operations are commutative. And then thirdly, we require that every element is ident potent with respect to both operations. So for any element x, we require that x join x is equal to x. We also require that x meet x is equal to x. So if we stop there for a moment, what we're saying here is that the set L join is a simulatis and L meet is a simulatis. So by themselves, each of these operations forms a simulatis. Now, if we just stop there, we have two simulatises that's great, but why study them together? Why not study them individually as a simulatis? Well, the point is that there's got to be some type of compatibility, some type of interaction between the two operations. With ring theory, we had the notion of distribution that multiplication distributes over addition. With simulatises, we don't require distribution, we actually require something weaker, the so-called absorption axiom. So the absorption action, because these axioms right here, this only involves join, this only involves meet, no interaction there whatsoever. So I need at least one axiom that requires meet and join together, otherwise it's not really a lattice, it's just two simulatises. Two halves should make a whole, how to meet and join and interact. So given any two elements x and y inside the lattice, we require that the join of x and x meet y is equal to x. And we require that the meet of x and x join y is also equal to x. So notice here that when you toggle these things, if you take x join x meet y, that's the same thing as x. And similarly, if you take x meet x join y, that's the same thing here as x. And you do have to be careful with parentheses here. We don't necessarily have an order of operations that because of the distributive law when we work in a ring, we give preference to multiplication over addition. Because the distributive law makes that thing unambiguous. In our context here, we don't have an order of operation. There's no please excuse my dear and Sally or anything like that. So we do have to put parentheses so we don't get confused. Do we do meet first or join? We don't have a preference. We do the same thing with sets with unions and intersections. There's no priority like we always do unions over intersections. We just like, oh, let's use parentheses. The same is also true here in a lattice. Now I want to give you some context to explain this absorption axiom. Why did we take this as our axiom here? So to explain absorption, think about the following, the least upper bound. So let's look at the first one right here because the join is going to give you the least upper bound. The least upper bound between X and X meet Y, which X meet Y is going to be the greatest lower bound here. So in particular, we have that X join Y is going to be less than or equal to X. Because X join Y is the greatest lower bound of X and Y. So in particular, it's less than X. So if I'm going to take the least upper bound between X and something less than X, that's going to be just X itself. Since X is bigger than this element, the least upper bound is X. So the absorption property is going to capture the fact that we want this partial order to be well behaved with regard to two operations here. And then going the other direction, the meat is trying to capture the greatest lower bound. Alright, so we're looking for an element that's less than X and less than X join Y. And we want the largest thing that's less than both of these things. But wait a second, by construction, X join Y should be the least upper bound between X and Y. So in particular, X is less than or equal to X join Y by construction of how this partial order is defined. And so if we want the largest thing that's less than X and less than X join Y, well since X is less than itself, well less than or equal to itself and it's less than or equal to X join Y, then the largest thing that's less than both of them is going to be X. So in terms of the partial order, these absorption axioms are quite natural and we need interactions between the two operations in order to have an effective algebraic theory right here. Alright, and so this proposition is going to basically capture this idea of ordering that I'm just talking about here. So given a lattice L join meat, then we say that the following three things are equivalent to each other. A join B is equal to B if and only if A is less than or equal to B if and only if A meet B is equal to A. So this proposition actually establishes the fact that A join B is the least upper bound of A and B and that A meet B is the greatest lower bound of A and B. So that's really what this proposition is doing. Now I want to be aware that the first one we've already taken care of because after all how do we define this operation? We say that A is less than B if and only if A, well basically it's just right here. This right here is the definition of what it means to be that, okay? And so that we took care of when we introduced simulatices earlier. So you can look to proposition 19-1-10 from this lecture series to get exactly the idea. This is the definition of A is less than or equal to B, so there's nothing to prove there. We already took care of it. But what about the other direction, right? A is less than or equal to B if and only if A meet B is equal to A. When I introduced the partial order on simulatices, I said you could get away with either one of these and this is the proposition that proves it. It doesn't really matter which direction you go. They're both equivalent to each other. So really we want to then convince ourselves that these two statements are equivalent to each other. A join B equals B if and only if A meet B is equal to A. And so let's do that. Let's suppose that A join B is equal to B. So then A join B is equal to A, excuse me, A join B is equal to B. Let's look at the product A meet B. So by assumption B is equal to A join B, so I'm going to substitute that in so we get that A meet A join B. But by the absorption axiom, this is equal to A. So we get that A join B is equal to A when A join B is equal to B. And the reverse direction is basically the same exact argument. Assume that A meet B equals A. And then consider A join B. Well by substitution we can replace the A with A meet B, like so. And now this time we do have, it's not exactly the same argument, but because of commutivity we can move the B to the front. So we get B join A meet B. And then also using commutivity here, A meet B, that becomes A meet, excuse me, B meet A. And then we can use absorption. So commutivity does is required to move some things around. But that's because when it came to absorption, we only assumed less left absorption. But because of the commutivity axioms, if we have left absorption, we also have right absorption. So we're good to go in that regard. So a couple of things to mention about this proposition here. So we define the partial order to be this equation. But because these two notions are equivalent, we could also get away with defining the partial order using that relation. That's why with sitting lattices, when you only have one or the other, it doesn't matter which direction you're pointing because you can get away with both of them in that situation. So in fact, the joint does give you least upper bounds and the meat does give you greatest lower bounds with regard to this partial order. But something else that's very important here, this proposition introduces a very important principle of lattices. That is the principle of duality. As the axioms of a lattice are identical. I mean, look at these things. I think of like Fred and George Weasley here. We're identical. They're exactly the same axiom that associativity holds for join and meet. Just replace the two symbols. Commutivity holds for join and meet. Idenpotency works for join and meet. And then if you take the absorption axiom over here, where you have join and then meet. If you replace the join with a meet and replace the meet with a join, you get the exact, the second axiom. And so this pattern works in general that whenever you have a statement about lattices where you have a join, if you replace that join with a meet and conversely replace every meet with a join, that gives you the so-called dual statement. The dual statement. You can replace each of the meets with a join and the joins with the meet. If that original statement involving joins and meets was true, then the dual statement involving meets and joins will also be true by the so-called dual proof. That as you take the exact same proof that you had before and replace all the joins with meets and all the meets with joins, you also might have some inequalities. You can replace less than or equal to with greater than or equal to, right? You have the dual statement. You'll get the dual proof and it'll still be valid. This then leads to this principle of duality. Any statement that is true for all lattices remains true whenever less than or equal to is replaced with greater than or equal to and meet and joins are interchanged throughout the statement. Any statement that's true for a lattice, its dual statement is also true by the dual proof. As such, you can basically get, every time you prove something for a lattice, you get two statements for the price of one because you prove something and then you get the dual statement for free because of this principle of duality. We'll see an example of that in just a second. I just wanted to bring it up now because in the previous proof, we basically did the exact same thing as we had the dual principle. The second proof was the same because of duality. Let's look at some examples. The poster child, the canonical example of a lattice is the lattice where you take the power set of some set and you equip to it unions and intersections where union takes on the role of join and intersection takes on the role of meet. After all, join looks like a union symbol that comes to a point and meet looks like a intersection symbol that comes to a point. We're just trying to generalize this structure right here. So for example, if you take a set with three elements, say ABC, then we can look at the haze diagram for its subsets. That is, this is the haze diagram for the power set, which has a natural ordering with respect to set containment. I mean, I want to point out that when you look at this picture, this haze diagram, the reason we call these things lattices is because the haze diagram, which is a visualization of the lattice structure looks like a lattice in the visual perspective. And some people call haze diagrams lattices. So a lattice is just the algebraic structure of a haze diagram. Now, not every haze diagram, not every partial order gives you a lattice. You do have to have well-defined least of our bounds and greater slower bounds. But if you do so, that's exactly what we have here as a lattice. So give us some examples here. If you take the singletons A and B, their union is AB. And we can see this in our diagram. If you take A and B, their least upper bound would be their union A and B. And going the other way around, if you take two sets, AB and BC. So indicate them on my lattice here. Here's AB and BC. Their intersection, which is B, coincides to their least, excuse me, their greatest lower bound on this diagram. And lattices are trying to generalize the structure we see right here. Now, it turns out this, there's more structure in play, right? Because when it comes to sets, we often think about set compliments, set differences and stuff. There's more structure going on there. This will naturally lead to what we call a Boolean algebra, which we'll define in a later video. But for now though, let's look at one more example of a lattice. I mentioned this earlier with simulatices. If we take any integer, any positive integer N, and you look at the set of positive divisors of that set, call that set bracket N. This forms a lattice where your operations are going to be the least common multiple is your join, and the greatest common divisor is your meet. And so, for example, when N equals 24, you see the following here. The divisors of 24 are 8, 12, 4, 6, 2, 3 and 1. If you take 12 and 8, their least common multiple is 24, and their greatest common divisor is 4 right there. We have least upper bounds and greatest lower bounds. We have a lattice structure here. If you take 2 and 12, their greatest, excuse me, their least upper bound will be 12, but their greatest lower bound will be 2 in that situation. The GCD of 2 and 12 is 2. The least common multiple is 12. If you take 4 and 6, their least common multiple is a 12. Their greatest common divisor is a 2. This structure, these binary operations of LCM and GCD form together a lattice. And that's exactly what we're trying to capture right here. All right, let me give you one last result about lattices in this video. We'll say more about lattices in lecture 38, of course. It turns out that the axiom of item potency is not necessary when it comes to a lattice. Now, I included in the original definition because I wanted you to realize that a lattice is two city lattices glued together using absorption. And with a city lattice, you have associativity, commutivity, and item potency. But because of the absorption axioms, you actually don't need item potency. That is to say the set L equipped with the two operations of joint and meet is a lattice if and only if these two operations meet and join satisfy the associative, commutivity, and absorption axioms. So the first direction is pretty obvious. If your operation satisfy associativity, item potency, excuse me, associativity, commutivity, item potency, and absorption, then clearly you satisfy the three. So that direction is easy. It's the other direction we need to focus on, that if we only have these three axioms, we can prove that elements are item potent with respect to meets and joins. Let's look at join first. So notice that if we take a join a, I claim that's equal to a join a meet a join a. That's really complicated, but how do we get away with that? Well, basically we go back to the absorption axiom. The absorption axiom says that x join x, excuse me, x meet x join y. This equals x for all x and all y. In this case, we're saying that both x and y are both equal to a. This is going to give us that a meet a join a is equal to a, and we make that substitution in for a right here. So we're going to get this statement right here. This is by the absorption of meet. We get this statement where all the elements are a in that situation. I didn't claim that this equals a, and we're going to use absorption in the other direction now, because the other direction for absorption tells us that x join x meet y. This equals x for all x and y inside the lattice. Now in this situation, I'm going to take x to equal a, and I'm going to take y to equal a join a in that situation. So notice you have x join x meet y by absorption that's equal to x. That's equal to a in this situation. So the absorption axiom gives us that a join a is equal to a. Wonderful. We use absorption with both meet and join, but we get that a join a is equal to a. And then I'm not going to prove the other one because of duality. The dual proof has to also be true if a join a equals a for all a inside the lattice, then the dual statement, excuse me, so we don't need that if there. So this is true for all a for all a there. This gives us the dual statement a meet a equals a for all a inside of lattice must also be true by the dual proof. And I don't have to prove it because of the principle of duality. It's so wonderful. It actually is pretty awesome. You know, all joking aside. And so that does bring us to the end of lecture 37 about lattices. We're going to develop some more theory of lattices in lecture 38, ultimately building up to the concept of a Boolean algebra, which is going to complete our analog here. Simi groups are two groups as lattices are two rings as Boolean algebras are two fields. So stay tuned for that one next time. If you learned anything about lattices or similasses in these videos for lecture 37, please like these videos, comment. If you have any questions, I'll be glad to answer them and subscribe to the channel to see more videos like this in the future. And I'll see you next time.