 Welcome back to our lecture series Math 42-30, abstract algebra two for students at Southern Utah University. As usual, be your professor today, Dr. Andrew Misseldine. So we're reaching basically the end of our lecture series. So previously in lectures, well, earlier in the 30s, we developed this idea of field theory, Gaoua theory culminated with the fundamental theorem of Gaoua. We competed some Gaoua groups, and then finally we proved the insolvability of the Quintic polynomial equation. Some pretty big notions here. So in these last three lectures for Math 42-30, what I want to do is basically do it all over again. But not in the complete depths we did before. So what do I talk about doing it all over again? If you look at both of these lecture series Math 42-20, abstract algebra one and 42-30, we studied in great depths the ideas of groups, rings, and fields. What I want to do is do it all over again, but basically now in an alternate universe, where instead of the group being the most fundamental algebraic notion for which we build on top of that, what is a ring? Well, a ring is just an abelian group with a second operation that behaves well with the first operation, which we'll call them addition and multiplication. And then a field is just a very highly structured ring. I want to do this over again, groups, rings, and fields, but in an alternate dimension, a bizarro world, the multiverse, so to speak, for which this time instead of developing the theory of groups, we're going to talk very briefly about the idea of a semi-lattice. In many ways, semi-lattices behave like groups, but in many other ways, they're quite alien objects. And then building upon that same theory, if we take a semi-lattice and add a second operation, we get the equivalent of a ring, what we call that a lattice. And then if we take a highly structured ring, what's like the perfect ring, well, that's going to give us a field as we did before, but in this context, we're going to lead to the idea of abelian algebra. And so very briefly, in the last few lectures of this lecture series, that's what we want to do, have a parallel between groups, rings, and fields in this setting of lattices. So the analog of a group in this alternate universe is what we call a semi-lattice. So a semi-lattice is a set equipped with a binary operation. So we take two elements, we combine them together and produce another element of that same set. It's going to have three axioms and some of the axioms are going to feel similar. So for a semi-lattice, our first axiom is going to be associativity. That's exactly what we took as the first axiom of groups. So just as a reminder, if we take three elements of our set, say the set is L, you take the elements X, Y, and Z, if you then take your binary operation, this, we could call this multiplication if we still wanted to, but I do want to make a look a little bit different than what we had from group theory. Also, I'm trying to generalize a different algebraic structure. What is the algebraic structure I'm trying to go through, right? With groups and rings and fields, we try to capture the notions of addition and multiplication. But as we study lattices in these lectures, we're not trying to capture the algebra of addition and multiplication. We're actually trying to go for the algebra of unions and intersections. I mean, at the very beginning of math 42-20, we spent time talking about the algebra sets. That is, we studied the algebraic structure of sets. So we have unions and intersections and set complements and things like that. And what we want to do now is generalize that concept to the algebra of order. Can we come up with an algebra unordered things, partially ordered sets, generally have algebraic structure and that's what we want to axiomize in this setting right here. Thus, with this semi lattice, our symbol here kind of looks like a union symbol, although it comes to a point. This symbol right here is commonly referred to as a join. A join sometimes it's referred to as a V. Like literally VV, you know, like V mon did evolves into V in that situation. I'll typically call it a join. That's a very common term. Sometimes people call it the least upper bound operator and we'll make some more sense of that by the end of this video. So we have our join operation and it's associative. So if you take Y join Z and then you join X onto that, that's the same thing as X join Y join Z. All right, so we do require this operation be associative. Our second axiom, slightly different approach to what we did with groups is our second axiom actually will be commutivity. So for any elements X and Y inside of the set L, we require that X join Y is equal to Y join X. We actually are gonna build commutivity into the cake here, cook it into the cake. For groups, that was sort of like, yeah, you could be commutative, you get an abelian group. But for simulatuses, it is required to be commutative. But again, that's not too alien from groups because we did study a lot about abelian groups. Commutivity was a good thing. It's our third axiom that really is gonna be deviant from what we expected with groups. The last axiom is idempotency. That is we require that every element be idempotent. So what that says in this context is if we have some element X inside of our set L, we have that X join X is equal to X. That is if you take an element and operate by itself, you always just get back the original element. Now this is a very alien concept from groups because in a group, the only idempotent element is the identity. And so in this respect though, we require for simulatuses, we require that every element is idempotent. And the poster child of a simulatus is exactly that of a power set with unions. So that is if you take any set whatsoever, consider it's power set, the set of all subsets, and you would join the power set with the union operator, this then forms a simulatus. Union is associative, it's commutative. And if you take any set, A union A, you just get back A. This gives you in fact a simulatus. We could also have done this with intersections. If you take the power set of X and adjoin to it the intersection operation, this is likewise a simulatus. And this is actually where the name simulatus comes from. It actually seems to suggest, I mean simi here kind of means like half half of a lattice with a lattice, a lattice which we'll define later on in this lecture, not in this video though. A lattice will be the structure that we get when we put these two things together. When we take unions and intersections together, it forms a lattice. So right now, if we only consider one of the operations, we get a simulatus, half of a lattice, all right? And so a simulatus is trying to generalize the structure of the union operation, much like how groups want to generalize addition and multiplication as algebraic operations. Simulatuses wanna generalize the notion of unions and intersections. But these are not the only type of simulatuses out there. For example, take the set bracket in to be, we're gonna take this to be the divisors, the, excuse me, the positive divisors. I want that to be the case here. We wanna have the positive divisors of some positive integer in like so. And then if you take this set in bracket and you equip it with the divisibility, the divisibility, excuse me, not divisibility. I'm getting a little ahead of myself there. Divisibility is not a binary operation. It doesn't produce an element, it's a relation. What I wanna do is actually, excuse me, the GCD, GCD in that situation. This also forms a simulatus for which, for example, you could take the divisors of 12. So you have 12, six, four. Let me scoot this up a little bit. We're gonna get three over here, two over here, one. Whoops, there's no line there. You get this picture right here. And so if you look at the divisors of 12, and you can take the operation of GCD. So like for example, if you take three and two, their GCD is one. If you take six and four, their GCD is two. If you take six and three, their GCD is three again. This gives you a binary operation, and GCDs will in fact be associative, it'll be commutative, it'll be idempotent. If you take the GCD of six and six, it's still six. Sorry, no, that's sort of like the devil's number there, 666, but it's just a mathematical example. We can create a simulatus using GCDs. Alternatively, you also could create a simulatus using LCMs, the least common multiple of these things. This would likewise be a simulatus. And this is to suggest why we call these things simulatuses. Like I said, simulatuses is half of a lattice. Why a lattice? Well, when you look at these Hase diagrams, which some people call Hase diagrams, lattices. The Hase diagram is basically the algebraic structure, or excuse me, a lattice is the algebraic structure of this type of partially ordered set. You can create a simulatus from a Hase diagram from a partially ordered set when you have a well-defined greatest common divisor or least common multiple. That is when you have a least of our bound or greatest lower bound on a partially ordered set. And that's actually what I wanna prove right here. This is the main result for this video on simulatuses is that the construction, the example I just mentioned is basically the only example of disomorphism. Which don't get me wrong, there's a lot of diversity when it comes to simulatuses. It's very easy to come up with them because we can just draw diagrams. We'll do that at the end of this video. But in the meanwhile, every simulatus has a natural partial order structure onto it. And therefore, the simulatus is capturing the algebraic structure of that partial order and vice versa. So imagine we have a simulatus, L with the symbol join, it's a simulatus. We can then define a partial order on L. That is we say that X is less than or equal to Y if and only if the product X join Y is equal to Y. This defines a partial order on the set L. So L is naturally a poset, partially ordered set. And in fact, the element X join Y is necessarily the least upper bound for X and Y with regard to this partial order. All right, so to prove it to partial order, there's three things we have to show. It has to be reflexive. It has to be anti-symmetric. It has to be transitive. So that's what we're gonna start with. So assume that L is a simulatus and consider the element X. Well, by the idempotency axiom we have that X join X is equal to X. Well, when we look at the definition of what it means to be less than, that's exactly it. X is gonna be less than or equal to itself. So this is in fact a reflexive axiom, a reflexive ordering, a reflexive relation. So in particular, idempotency directly implies that this thing is reflexive. I want you to pay attention to these things. Three axioms of simulatus, three axioms of the partial order. So let's prove that it's anti-symmetric. Take elements X and Y inside of the lattice, simulatus, and let's suppose that X is less than Y and Y is less than or equal to X. Well, if X is less than or equal to Y, that means that X times Y is equal to Y and if Y is less than or equal to X, that means Y times X is equal to X. But this is a commutative operation. So X by this assumption is equal to Y times X, but since it's commutative, this product is equal to this product X times Y, but X join Y is equal to Y by this assumption and so we get that X equals Y and so our relationship is anti-symmetric. So this time we see that the commutivity axiom implies the anti-symmetry axiom. So this is one-to-one correspondence between axioms coming into play here. So maybe it comes for no surprise here that the transitivity property of the relation will follow exactly from associativity. That's what we're gonna see here. So let's suppose we have three elements, X, Y, and Z that belong to the simulatus and suppose that X is less than or equal to Y and Y is less than or equal to Z. Well, since X is less than or equal to Y, that means that X times Y is equal to Y and since Y is less than or equal to Z, that means Y times Z is equal to Z. Now we're gonna use associativity here. So what we need to show is that X join Z is equal to Z. I would show that X is less than or equal to Z. Well, X join Z, because Z is equal to Y join Z, we can substitute that into the equation. So we get that X join Y join Z is then equal to X join Z. But then by associativity here, aha, associativity is gonna give us this equality right here. X times YZ is equal to XY times Z. But then by the previous assumption, XY is equal to Y, we make that substitution. And so we get that XY times Z is equal to Y times Z. But then also by previous assumptions, Y times Z is equal to Z. And so that then gives it to us. X times Z is equal to Z, so therefore X is less than or equal to Z. This is a direct result because of the associativity axiom. So we get transitivity. There's this direct relationship here. Idenpotency, implied reflexivity. Commutivity implied anti-symmetry and associativity implies transitivity. So we get this partially ordered set here. Every simulatus has a natural partially ordered set relation. Now I do wanna make a comment here that the way we defined it, we said that X is less than or equal to Y if and only if X join Y is equal to Y. This is not how everyone works with simulatus. Some people go the other way around. Some people will say alternatively that X is less than or equal to Y if and only if X union Y equals X. So they go the other way around. Compared to what we've developed here, this would be the reverse relation, the reverse partial order. Given any partial order on set, you can always turn them around. And so for our purposes, this is what we would say is greater than or equal to. So when it comes to simulatus and lattices, sometimes they have the orders flip around. This is particularly the case with simulatuses. So don't lose too much sleep over such a thing. That is the case. Now, personally for me, when you define it like this, you use the symbol join to describe it. In this setting, really what you should say is that X is greater than or equal to Y if and only if X, well, let me back up. If you wanna say it the way that I had it written before, X is less than or equal to Y. In that setting, you'd actually say that X join Y is equal to X. Sorry, not join. This symbol right here is what we call the meat. Or some people call it, what else do people call it? Sometimes they call it a smash. We'll stick with meat in this situation. Wedge, that's a term that some people use here, a wedge. That was the word I'm looking for. Some people actually you call this symbol wedge. So you have to be very careful. But in latex, this symbol right here is backslash V. V-mon, digi-volve two, like I said before. And this symbol is the wedge symbol in latex. So some people call them that as well. We will typically call these meat and join. This is join, this is meat. When it comes to a similatus, there's really no distinction between the two. The two can be equivalent to each other. But when it comes to a lattice, we will distinguish between what a meat is and what a join is because there's a harmony that lives there. All right, so continuing on. Given the definition of the partial order that we have, we wanna show that the symbol X join Y is the least upper bound with respect to this order. That is to say, to show that it's the least upper bound, we have to show that X and Y are less than X, X join Y. That's not too hard to see here because if I take X join X Y, because it's associative, you end up with X join X for which, since it's idempotent, you get X join Y. So this would give us that X is less than or equal to X join Y. And similarly, you're gonna get Y is less than or equal to X join Y. You might have to use commutivity in there, not a big deal, but all the accents come to play. So this tells us that X join Y is an upper bound of X and Y for which if you slightly define the partial order like we were doing here, this would actually give you a lower bound, but we'll stick with what we have here. Y is at the least upper bound. We'll take any other element, Z, in that situation. Suppose that X and Y are both less than or equal to Z, right? So that means that X join Z is equal to Z and Y join Z is equal to Z. So what about X join Y? Well, if you join that with Z, by associativity, this will be X join Y Z, but by assumption, Y Z is just a Z, and then by assumption, again, X Z is equal to Z in that situation. So this shows us that X join Z is less than or equal to Z and since Z was an upper bound and X join Y is less than every upper bound, that makes it the least upper bound. And so we see in this situation that given a semi lattice, you can put onto it a partially ordered set for which you always have least upper bounds. The least upper bound is in fact the quote unquote product of these things, the join of these things. Now, it turns out that this process is reversible. That is, if you start off with a set, if you start off with a set X and you have some partial order on it, such that least upper bounds always exist with respect to this partial order, then it turns out you can actually create a semi lattice by reversing this process. Since the least upper bound exists, you say that X join Y is equal to their least upper bound. You can then argue that this operation is associative, it's commutative and it is idempotent, thus giving you a semi lattice. Although I'm gonna leave that as an exercise to the viewer here.