 i is a linear order, then implies that gi is a subgroup of gj. So it's a chain of groups. We can think of i as being the natural numbers, and the group's an exponential number for first approximation. Then you can take the limit or the union of this chain. And this will already be a group. Then, so you might wonder what I mean by the union. So let me tell you what it is in details. So what is g exactly? So the universe of g, the set of elements of g, is just going to be the union of the elements of the gi, so this is easy. Yeah, so when I say gi is a subgroup of gj, I mean that's the one of gi, the same as the one of gj. And the operations match up, and the operation matches up. So products, so if a and b are in gi, then a times in the sense of gi, b, is the same as a times b in the sense of gj. So multiplication in gj restricted to gi. So this is the universe, and how you're going to define the binary operation of g is just going to be what you think. So the only possible thing. So if I take a, b in g, then a times b in the sense of g would be defined as a times b in the sense of gi, where i is big enough to contain both a and b. And there are many possible i's like this, but by this property, it doesn't matter which one it is. So you can always define the union, but the important part is that it's a group. And why is it? Well, basically, I mean just check. So what we have to check that the operation is associated, it's fine, that inverses exist, it's all fine. If you take a in g, then it's in gi, so it's an inverse in gi also. And that's basically it. But that's an important property of groups. And many other algebraic objects have this property. Exercising undergraduate algebra for your favorite thing. So I don't know. If you take fields, they have this property. Vector spaces. So what I want to do is give a framework where basically it axiomatizes this property and nothing more, really. And then still see that we can still say something interesting about a nice class in that framework. But unfortunately, I have to start with some background and logic to make sense of it. So what I want to start with is generalize the notion of a group to the notion of a structure. So a structure will just be a universe together with a bunch of operations. And I need to specify what the operations will be. So if this is background, the definition of language is a set consisting of functions symbols and constant symbols. And I would say what I mean exactly. And to each of these things, each function and relation symbol, it also comes in an arithmic natural number. So for example, the language of group has this binary operation. So it has a symbol for this binary operation times. And it has also a couple more symbols. This is going to be a definition, but hopefully you can see how it relates. So it has a binary function symbol, I guess. It's a time operation. I'm also going to have a fixed constant symbol for the unit. And also a unary function symbol for taking the universe. So I could have omitted some of these things, the last two, but it's going to be important that I include the operation of taking the universe. So language is just a bunch of symbols. It doesn't tell me anything about how the operation behave. It just tells me, well, this is what I have to work with when I want to talk about groups. What do I do with the language? I can define n structures or models. So in that case, the language of group has no relation symbols. You could imagine that I'm putting an ordering on the groups. So if you put your order group, it would be a relation symbol less than a binary relation symbol. So let me give you another example. The language of linear orders. So that's a different language. I will define it just as having one binary relation So when you talk about linear order, you just have this ordering less than, and that's just a binary relation. But groups, well, basically, they only have functions. You don't usually have, in general, you don't define any relation. You define relations eventually. In the basic language of groups, you just have functions. Hopefully it will come clearer. Let me define what an n structure is. So for add a language, it has a universe. So it's an omitted set. So for a group, it would just be the set of elements in the group. So I would call this a set a, so that I can refer to it. And then, what else do you have to specify when you study groups? You have to say what the multiplication symbol is. And in general, what all the other symbols are. So when you study the orders, you would have to specify what the ordering is. So for each symbol in the language, you have to specify an interpretation, a way to make sense of how the function will act on the universe. So for each, so I will do it for function symbols. In symbol f in the language, let's say it has arity n. I will specify some interpretation, fm, which will send n tuples in a to an element. So in the case of group, this would be a multiplication function n would be equal to 2. And similarly for the others. So similarly for constant symbols and relation symbols. So this is just going to tell me which vocabulary I have to work with, just so that I don't mix up models that are linear orders with models that are groups. But then another point that I will make clear very soon is that this too doesn't tell me what the group is. It just tells me what is a structure with a binary operation. But then it doesn't tell me anything about the axioms of group. So let me give you two examples. So first, so just work in the language of group. So let's m1, or let's m be the n structure. So universe will take it to be z. So this is just a definition for the person of illustration. And then multiplication would be addition. So a and b and a, I have to define. Since I want to define a structure, I want to define interpretation for each of these things in the language of groups. So I will define what it means to multiply a by b. So OK, so here I've written the function in this authentic way. But you'll usually write multiplication like this and you won't write times a, b, or something like this. So I'll write it this way. So I'll just define this to be a plus b. And then similarly, I define inverse and units. So the 1 would be the 0 and the inverses. So then this is just the integers with additions. This is a group.