Linear Orders
Meow meow :3
Maybe I'll create a pfp later ^^ But for now, linear orders!
Today, I'll be talking- typing about linear orders, I'll abbreviate this to simply LO. These are mathematical structures that look like you can put them on a line :3
Here's what we'll do today:
In the introduction, I'll explain what a linear order is and I'll explain what ω, ζ, η and θ are.
In chapter II, I'll explain the category of linear orders: what morphisms are, what embeddings are, and I'll define a relation ≼ on LO's.
We'll look at arithmetic of linear orders and show some basic facts about them.
In chapter IV, we'll take a closer look at η and explore dense orders.
In chapter II part II, a.k.a chapter V, We'll look at automorphisms of some LO's. In particular, we'll take a closer look at Aut(ζ).
Ordinal numbers! :D
In this chapter, we'll look at the topology of LO's.
We'll end with connected orders. This hopefully completes the basic picture of the LO's introduced in chapter I.
I. Introduction
A linear order is a structure (A,≤) with a set A and a binary relation ≤ on A such that:
≤ is reflexive: x ≤ x for all x;
≤ is transitive: if x ≤ y and y ≤ z, then x ≤ z;
≤ is antisymmetric: if x ≤ y and y ≤ x, then x = y;
≤ is total: x ≤ y or y ≤ x for all x and y.
Intuitively, you can put all points of a linear order on a line, and one point x is less than another point y if it's to the left of it.
Let's look at some examples! For every finite number n, there is exactly one (up to isomorphism) linear order with n points. ω is the order type of ℕ. (An order type is basically what a structure looks like when only looking at the order.) ζ is the LO of ℤ, η is the order of ℚ and θ is that of ℝ. No one can really agree on what letter to use for ot(ℝ), I've also seen λ and ρ used, but I'll use θ throughout this blog-post. Here is a fun picture depicting these:
I call a linear order left-bounded if it has a smallest element, right-bounded if it has a greatest element, left-unbounded if it has no smallest element, right-unbounded if it has no greatest element, bounded if it's both left- and right-bounded and unbounded if it's both left- and right-unbounded.
We can see that 0 is unbounded, 1 is bounded, ω is left-bounded (0 is the smallest element) but right-unbounded (for every n ∈ ω, we have n + 1 > n) and ζ, η and θ are all unbounded.
Linear orders of any size exist. ω, ζ and η are all countable, meaning that we can enumerate the points of them in a list. However, θ is uncountable. You can read my blog post about cardinal numbers if you want to understand infinite sizes better.
II. Morphisms
A morphism from a linear order (A,≤A) to a linear order (B,≤B) is a function f: A → B such that, for all x,y ∈ A, if x ≤A y, then f(x) ≤B f(y). You can think of a morphism as a function that moves the points around, but never "swaps" the order of two points. Though it may not swap two points, it can put two points in the same place. For example, n ↦ ⌊n/2⌋ is a morphism from ω to itself.
In this category, 0 is the initial object and 1 is the terminal object. This means that, for any α, there is a unique morphism from 0 to α (the empty function), and a unique morphism from α to 1 (it sends every x ∈ α to the unique element of 1).
In category theory, a monomorphism is a morphism f: A → B such that, for any object C and any two g₁,g₂: C → A, if f ○ g₁ = f ○ g₂, then g₁ = g₂. This might seem complicated, but in the category of linear orders, this just means that f is injective. Equivalently, x ≤ y if and only if f(x) ≤ f(y). I'll call monomorphisms embeddings from now on. Embeddings are a way one LO can sit inside another. I'll write f: α ↪ β to mean that f is an embedding from α into β.
An epimorphism is a morphism f: A → B such that, for any object C and any two g₁,g₂: B → C, if g₁ ○ f = g₂ ○ f, then g₁ = g₂. In LO, epimorphisms are exactly the surjective morphisms. Thus, f: α → β is an epimorphism if the image of f is β. I'll write f: α ↠ β to mean that f is an epimorphism.
An isomorphism is a morphism f: A → B for which there exists an inverse morphism f⁻¹: B → A s.t. f⁻¹ ○ f = id_A is the identity morphism on A and f ○ f⁻¹ = id_B is the identity morphism on B. In LO, this means that f is bijective. I'll write f: α ≅ β to mean that f is an isomorphism.
If there exists an isomorphism between α and β, then α and β are isomorphic. I'll treat isomorphic linear orders as the same linear order. Thus, I'll write α = β for ‘α and β are isomorphic’.
I'll write α ≼ β (‘α embeds into β’) to mean that there exists an embedding j: α ↪ β. We can see that ≼ is a pre-order:
≼ is reflexive: α ≼ α for all LO's α;
≼ is transitive: if α ≼ β and β ≼ γ, then α ≼ γ.
However, it is not antisymmetric or total. Try to find counterexamples to this! I.e., try to find some α and β so that α and β are not isomorphic, yet α embeds into β (α ≼ β) and β embeds into α (β ≼ α). And try to find γ and δ such that neither γ embeds into δ (γ ⋠ δ) nor δ embeds into γ (δ ⋠ γ).
Since ≼ is not antisymmetric, we can have α and β such that α ≼ β and β ≼ α, yet α ≠ β. α and β that embed into each other I'll call order equivalent, denoted α ≡ β. This means that they're sort-of equal, but not really.
If α ≼ β and β ⋠ α, then I'll write α ≺ β (this is not the same as α ≼ b ∧ α ≠ β). We have ω ≺ ζ ≺ η ≺ θ.
We'll look more at isomorphisms and automorphisms (isomorphisms f: α ≅ α from an object to itself) in chapter V.
III. Arithmetic
Mrrowr :3
In this chapter, we'll look at the three basic operators *, + and × on linear orders. We'll start with the simplest one, *!
For a linear order α, α* is the dual order of α. α* has the same points as α, but the order is reversed: x ≤ y is true in α* iff y ≤ x is true in α.
We can see that if we reverse the order of any finite LO n, we'll just get n back. I.e. n* = n. Some infinite α are also equal to its dual, e.g. ζ* = ζ, η* = η and θ* = θ.
If we take the dual of the dual (thus, we flip α twice), we just get the same LO back. I.e. α** = α.
For two linear orders α = (A,≤A) and β = (B,≤B), we can add them together to create a new linear order (A,≤A) + (B,≤B) = (A+B,≤). A+B is the disjoint union of A and B, meaning that points in α+β are of the form (a,0) and (b,1) for a ∈ A and b ∈ B. We have the usual order of α and β in α+β: (x,0) ≤ (y,0) iff x ≤A y and (x,1) ≤ (y,1) iff x ≤B y. In α+β, everything in A is to the left of everything in β, thus (x,0) ≤ (y,1) for all x ∈ A and all y ∈ B.
You can view α+β as taking α and adjoining β to the right of it (or taking β and adjoining α to the left of it).
Here are some basic facts about addition:
0 is the identity for addition, i.e. α+0 = 0+α = α;
Addition is associative, i.e. (α+β)+γ = α+(β+γ) = α+β+γ;
α ≼ α+β and β ≼ α+β.
However, as it turns out, addition is not commutative! OwO Try to find α and β for which α+β ≠ β+α!
We can see that ζ = ω* + ω, 6+ω = ω, η+η = η and θ+θ ≠ θ, but θ+1+θ = θ.
We can see that addition interacts with duality in an interesting way: (α+β)* = β*+α*. Thus, taking the dual of a sum is the same as summing up the duals, but in reverse order :P.
The most complicated basic operation on linear orders is multiplication. For linear orders α = (A,≤A) and β = (B,≤B), points in αβ are pairs (a,b) of a point a in A and a point b in B. In αβ, (a,b) ≤ (c,d) iff b < d or [b = d and a ≤ c]. Intuitively, you take the order β and replace each point with a copy of α.
Multiplication is associative, (αβ)γ = α(βγ), * distributes over multiplication, (αβ)* = α* · β*, and multiplication is left-distributive over addition, α(β+γ) = αβ+αγ. Of course, you can try proving these basic facts if you want to. Just like addition, multiplication isn't commutative. Finding α and β for which αβ ≠ βα is left as an exercise. Here is something funny: although multiplication is left-distributive over addition, it isn't right distributive! Thus, for some α, β and γ, we have (α+β)γ ≠ αγ+βγ.
I'll often write α^n for α multiplied with itself n times. It isn't really possible to exponentiate with infinite linear order powers. If we have linear orders α and β, where β is infinite, we need some "center" or "zero" 0 ∈ α if we want to define α^β. If we have chosen such a 0, we can define α^β to be the order-type of finite support functions f from β to α, where ‘finite support’ means that {x ∈ β | f(x) ≠ 0} is finite ({x ∈ β | f(x) ≠ 0} is called the support of f). If we don't require f to have finite support, then lexicographical ordering might not be possible.
I'll stop talking about exponentiation and centers of linear orders now, so you can explore more of this on your own. There might also be different ways to exponentiate linear orders.
IV. Dense Orders
[Definition] A linear order α is dense iff for all x,y ∈ α, if x < y, then there is some z ∈ α so that x < z < y.
Thus, between any two points, there is another point. A dense order can alternatively be defined as an order in which every point is a limit point, we'll talk more about limit points and discrete orders in chapter VII.
Trivial examples of dense orders are 0 and 1. These are dense because there aren't enough points for them to have x < y somewhere, so they're vacuously dense. I'll call an order that isn't 0 or 1 a non-trivial linear order. Any finite LO beyond that (2, 3, 4, etc) isn't dense. ω and ζ also both aren't dense, while η and θ are dense. θ is a bit more than dense: it is connected, which I'll talk more about in chapter VIII.
In some way, η is the simplest (non-trivial) dense order, as it embeds into every other dense linear order. Simultaneously, it is the most complex countable order, as every countable order embeds into it. Both follow from the theorem below:
[Theorem] Every countable linear order embeds into every non-trivial dense linear order.
Please try to prove this theorem yourself before reading my proof.
[Proof] Let α be a countable linear order and let β be a dense linear order. And assume, without loss of generality, that β is unbounded: if β is left- and/or right-bounded, then we can simply cut off the ends, making it unbounded by our assumption that it is dense. By the assumption that α is countable, we have some enumeration a₀,a₁,a₂,... of points in α. We can define an embedding f: α ↪ β by induction. Basically, we put more and more points from α in β, making sure each time that they're in the right spot. First, let f(a₀) be any point in β. Suppose f(aₘ) is already defined for all m < n, we'll now define f(aₙ). We have the set L = {f(aₘ) | m < n; aₘ < aₙ} of points to the left of f(aₙ) (or, well, where f(aₙ) should be) and R = {f(aₘ) | m < n; aₘ > aₙ} of points to the right of where f(aₙ) should be. Since L is a finite set, it must have some maximal element l = max(L). And since R is finite as well, it has some minimal element r = min(R). If L is empty (and thus, l does not exist), we can take f(aₙ) to be some number below r, which exists as β is left-unbounded. Dually, if R is empty, we can take f(aₙ) > l. If both l and r exist, we can take f(aₙ) to be some point such that l < f(aₙ) < r, which exists as β is a dense order. ∎
Since η is countable, it embeds into every non-trivial dense order, and since η is dense, every countable order embeds into it. We thus have that all countably infinite dense orders are order equivalent. It turns out that η ≡ η+1 ≡ 1+η ≡ 1+η+1 are the only countably infinite dense linear orders, I leave a proof of this as an exercise to the reader.
I'll end this chapter with a list facts about how dense orders interact with arithmetic:
α is dense iff α* is dense.
α+β is dense iff α is dense, β is dense and at least one of the following holds: α is right-unbounded, or β is left-unbounded.
αβ is dense iff α is dense and [α is not bounded or β is dense].
These are all pretty easy exercises.
V. Automorphisms
In group theory, a group is a mathematical structure (G,·) with a set G and a binary operator · such that:
There is an identity element e ∈ G: e·x = x·e for all x ∈ G;
Every x ∈ G has a unique inverse x⁻¹ ∈ G, x·x⁻¹ = x⁻¹·x = e;
· is associative, i.e. (x·y)·z = x·(y·z) for all x,y,z ∈ G.
One type of group is an automorphism group. Given an object A, the automorphism group of A, denoted Aut(A), is the set of all automorphisms f: A ≅ A. In this group, we take morphism composition (written ○) as our binary operator. In this group, the identity element is the identity morphism and the inverse element is the inverse morphism.
The trivial group is the group with a single element, which is the identity element. I'll write the trivial group with a bold 1. Some linear orders have the trivial group as automorphism group, for example Aut(2) = Aut(ω) = Aut(ω2) = 1. There is no way to move the elements of ω around other then leaving them all where they started.
Some linear orders have a more interesting automorphism group. For example, Aut(ζ) = (ℤ,+) (the cyclic group of order infinity) and Aut(η) and Aut(θ) are kinda complicated.
To explain why Aut(ζ) = (ℤ,+): an automorphism of ζ shifts the elements to the left or right by some amount x. First shifting by x amount and then shifting by y is the same as shifting by x+y. We thus have that the automorphism group of ζ is the integers under addition.
The automorphism group of θ corresponds to strictly increasing continuous functions on the real number line. It has 𝔠 many elements. I don't know if this group has been researched a lot, tell me if you find anything interesting about it!
One natural question to ask is: what groups can be the automorphism group of a linear order?
I'll give you part of the answer to this question. A subgroup of a group (G,·) is a set H ⊂ G such that (H,·) is itself a group: the identity element of G must be in H, the inverse element of any x ∈ H must be in H and, for any two x,y ∈ H, we have x·y ∈ H. Given a set X ⊂ G, we write ⟨X⟩ for the subgroup of G generated by X. This is the smallest subgroup of G that includes X. Given a single element a ∈ G, we can also have ⟨a⟩, which is the smallest subgroup of G that contains a. If a = e is the identity element, then ⟨e⟩ = {e} is just the trivial subgroup. For other a, we have ⟨a⟩ = {..., a⁻², a⁻¹, e, a, a², ...}. In group theory, we often write a^n for a · ... · a w/ n copies of a. We might have something like a³ = e, in which case, {..., a⁻², a⁻¹, e, a, a², ...} = {e, a, a²}. However, we can also have a, a², a³, a⁴, etc, be all different elements of G. In which case, we have ⟨a⟩ ≅ (ℤ,+)
It turns out that, in the automorphism groups of linear orders, if f ∈ Aut(α) is an automorphism that is not the identity, then ⟨f⟩ must be isomorphic to (ℤ,+). We can see this pretty easily: if f moves some x ∈ α to the right, i.e. f(x) > x, then it must also move f(x) to the right, and f(f(x)) = f²(x), and f³(x), etc. Meaning that f(..f(x)..), no matter how many applications of f you have, can never be x again. Thus, f, f², f³, f⁴, etc, must all be different automorphisms. This is only one restriction groups induced by linear orders must have, and I'm sure you can find more.
(ℤ,+) is in some sense the simplest non-trivial group that can be induced by a linear order. There are a lot of linear orders that induce (ℤ,+) (that have (ℤ,+) as automorphism groups). As mentioned above, Aut(ζ) = (ℤ,+), but this is also the automorphism group of ω+ζ, ζ+2, etc.
In the same way that η is the simplest dense LO, ζ is the simplest order with a non-trivial automorphism group:
[Theorem] ζ embeds into every linear order with a non-trivial automogrphism group.
Unlike η, where η+ω, 1+η+1, etc, also all embed into all dense LO's (and all dense LO's embed into them), ζ is the unique simplest linear order with a non-trivial automorphism group:
[Theorem] If α has a non-trivial automorphis group and embeds into every linear order with a non-trivial automorphism group, then α = ζ.
Another fun fact: we know when a linear order has a non-trivial automorphism group when ζ embeds into that LO.
[Theorem] ζ embeds into α iff α has a non-trivial automorphism group.
Proofs of these theorems are left as an exercise.
VI. Ordinal Numbers
In mathematics, a well order is a specific kind of linear order. A LO (A,≤) is defined to be a well-order if:
For all non-empty S ⊂ A, S has some minimal element x, i.e. for all y in S, we have x ≤ y.
Every finite LO n is a well order. ω is a well order as well but, e.g., ζ is not well-ordered: ℤ⁻ ⊂ ζ, the set of negative integers, does not have a least element. Order types of well orders are called ordinals. They are an important concept in set theory as they describe the heights of trees and sets, and because of transfinite induction.
[Theorem] For a linear order α, the following are equivalent:
α is an ordinal;
every strictly decreasing sequence in α is finite;
ω* does not embed into α.
[Definition] A set X ⊂ α is inductive if for all x ∈ α, if for all y < x, we have y ∈ X, then we have x ∈ X as well.
Ex. The set of all rational numbers below the square root of 2 is inductive in η.
[Theorem] If α is an ordinal and X ⊂ α is inductive, then X = α.
Both of these theorems are left as an exercise to the reader.
Ordinals have a lot of nice properties. For example, α+β and αβ for any two ordinals α and β are ordinals as well. Also, every infinite ordinal has a smallest element, which we can take as our center in exponentiation, meaning that ordinal exponentiation is well-defined. We also have that ≼ is itself a well-order on ordinals:
≼ is antisymmetric on ordinals: if α and β are ordinals, α ≼ β and β ≼ α, then α = β;
≼ is total on ordinals: for ordinals α and β, we have α ≼ β or β ≼ α;
≼ is well-founded: all sets of ordinals have a ≼-minimal element.
This means that the theorem of induction (X is inductive → X = α) also applies to Ord, the class of ordinals. We can also view each point in an ordinal as its own ordinal: for x ∈ α, we can define (x) = {y ∈ α | y < x}, and this set with the usual order of α is an ordinal (x) < α.
A von Neumann ordinal is a specific representation of an ordinal. It is a transitive set of transitive sets. For von Neumann ordinals α and β, α < β is defined as α ∈ β. Von Neumann ordinals are often used in set theory.
Given a set of ordinals S, the supremum of S, written sup(S), is the smallest ordinal α so that β ≤ α for all β ∈ S.
Here are some more examples of ordinals:
ε₀ (epsilon-nought) is defined as the smallest ordinal for which ε₀ = ω^ε₀;
ω₁ck (Church-Kleene ordinal) is defined as the smallest ordinal for which there is no Turing machine that defines an order that is isomorphic to ω₁ck;
ω₁ is defined as the smallest uncountable ordinal.
[Definition] Given a linear order α and a set S ⊂ α, S is cofinal in α iff for all x ∈ α, there is some y ∈ S so that x ≤ y. The cofinality of α, written cof(α) or cf(α), is the smallest cardinality (i.e. size) of a cofinal subset S ⊂ α.
For example, cf(0) = 0, cf(1) = cf(α+1) = 1 and cf(ω) = cf(ζ) = cf(η) = cf(θ) = ℵ₀. ω₁ has uncountable cofinality (it has cofinality ℵ₁), meaning that, for all countable subsets S ⊂ ω₁, there is some y ∈ ω₁ so that x < y for all x ∈ S. If it'd've'd countable cofinality, then we could take some countable cofinal S ⊂ ω₁ and some enumeration aₙ of S. Then, because ω₁ is the smallest uncountable ordinal, each ordinal (aₙ) must be countable. Thus, we can take injections fₙ: (aₙ) → ℵ₀. But then we can define an injection g: ω₁ → ℵ₀² by setting g(x) = (fₙ(x),n) for the smallest n for which x < aₙ. However, everyone knows that ℵ₀² = ℵ₀, so we have an injection g: ω₁ → ℵ₀ witnessing ω₁ is countable, thus a contradiction.
Please tell me if that was too hard to follow...
I'm going to sleep now. I'll write the next chapters tomorrow.
VII. Topology
It's midnight. Technically the next day.
In maphs, a topology is defined as a structure (X,τ) with a set of points X and a family τ ⊂ P(X) of subsets of X, such that:
The union of any number of sets in τ is in τ;
The intersection of any finite number of sets in τ is in τ.
The empty union is the empty set and the empty intersection is the full space X itself, so ∅ and X must both be in τ. In a topology, members of τ are called open sets. A set is closed if its complement is open. It is clopen if it's both open and closed. We can see that ∅ and X are always clopen.
Every linear order has an order topology. Given a linear order α and some point x ∈ α, (-∞,x) = {y ∈ α | y < x} is the set of points below x and (x,∞) = {y ∈ α | x < y} is the set of points above x. For x,z ∈ α with x < z, (x,z) = (x,∞) ∩ (-∞,z) = {y ∈ α | x < y < z} is the set of points between x and z. (x,z) is called the open interval from x to z. A set in the order topology on α is open iff it is a union of open intervals. You can verify that this indeed defines a topology. Another equivalent definition is: O ⊂ α is open iff ∀y ∈ O ∃x,z ∈ α ∪ {-∞,∞} x < z ∧ (x,z) ⊂ O.
A topological space in which every set is an open set is called a discrete space. A discrete space can alternatively be defined as a topology where every singleton (every set with a single element) is open. A limit point is a point x ∈ X for which {x} is not open. A discrete space thus is a space with no limit points. A limit point in a linear order α is a point x for which (1) for all y < x, there is some z < x so that y < z, or (2) for all y > x, there is some z > x so that z < y, the reader may verify that this is correct.
The order topology of any finite LO is discrete. ω and ζ both also have a discrete topology. However, the topology of η and θ are not discrete. In fact, η and θ are dense linear orders: all points in η and θ are limit points. ω+1 is neither discrete nor dense: it only has one limit point. (Assuming the axiom of choice) a discrete linear order of any size exists, a proof of this is left as an exercise to the reader.
In topology, a dense set (not to be confused with dense orders) is some set D for which, for all non-empty open O, D ∩ O is non-empty. For example, the set of rationals is dense in θ, while the set of integers is not (it does not intersect the open set (2.6, 2.74) ∪ (12.2, 12.2002)). Equivalently, D is dense iff D¯, the closure of D, is the whole space X. The closure of a set A is defined as the (inclussion-)smallest closed set that includes A, i.e. A¯ = ⋂{C | A ⊂ C ∧ C is closed} = {x | ∀O O is open ∧ A ⊂ O → x ∈ O}. I'll say a LO α is dense in another LO β if there is an embedding f: α ↪ β for which its range is dense in β. For example, η is dense in θ but not in θω₁. Usually, the bar is placed on top of the set A to denote its closure, but I can't do that here, so I'll write it next to it instead :P
For a set A, the interior of A, denoted int(A), is the largest open set included in A. I.e. int(A) = ⋃{O | O ⊂ A ∧ O is open} = {x | ∃O O is open ∧ O ⊂ A ∧ x ∈ O}. int(A^c) = (A^c)¯.
In topology, we often talk about local properties of a space. Given some point x in a topological space, a neighboorhood of x is a set U that includes an open set that contains x. Thus, x ∈ O ⊂ U for some open O. Given some A ⊂ X, we can define a topology on A as follows: τ_A = {O ∩ A | O ∈ τ_X}. (A,τ_A) is a subspace of (X,τ_X). A topological space (X,τ) has a property P locally iff for every point x ∈ X, there is some neighboorhood U of x with that property P. For example, αβ is locally isomorphic to α and ω is locally compact.
For two points x and y in a topological space (X,τ), I say x and y are connected if there is no clopen set A so that x ∈ A and y ∉ A. Equivalently, there are no open U and V such that x ∈ U, y ∈ V, U ∩ V = ∅ and U ∪ V = X. Trivially, every point is connected to itself. A topology is connected if all points in the topology are pairwise connected. It is completely disconnected if no two distinct points are connected. In a linear order α, two points x < y are connected if there is no "gap" between x and y. η is completely disconnected, as between any two rational numbers, there is an irrational. However, θ is connected. The reader can verify that a linear order is locally connected iff every limit point is connected to some other point.
A topology is compact if every open cover (that is, every family of open sets F such that ⋃F is the whole space) has a finite subcover (some finite F₀ ⊂ F that still covers the space). Every finite space is compact as the only open covers are already finite. ω + ω* is not compact as {{x} | x ∈ ω + ω*} is an open cover with no finite subcover, but ω + 1 + ω* is compact as any open set that includes the middle element (which I'll call ‘X’) must also include (n,X) and (X,m) for some finite n and m. Intuitively, a compact space is bounded and has no small gaps.
VIII. Connected Orders
Intuitively, a connected order is a linear order with no gaps or holes. Thus, θ is connected, but η is not as every irrational number forms a hole. In other words, it is very dense (the densetest it can be). 0 and 1 are trivial connected orders. The simplest non-trivial case is θ, as it embeds into every other connected linear order.
[Theorem] If α is a non-trivial connected LO, then θ ≼ α.
θ+1, 1+θ and 1+θ+1 are also connected linear orders with this property (thus, like with the η case, we have θ ≡ 1+θ ≡ θ+1 ≡ 1+θ+1). θ+1+θ = θ, however, θ2 ≠ θ as θ2 has a disconnect between the first and second copy of θ.
Not every non-trivial connected LO is order-equivalent to θ. For example, the long line, (1+θ)ω₁, is a connected linear order that is too long to be squashed into θ. The reader can verify that (1+θ)ω₁ ⋠ θ. Sometimes, (1+θ)ω₁ is referred to as the right side of the long line, ((1+θ)ω₁)* = (θ+1)ω₁* is the left side and the full long line is made by gluing the left and right side together, removing the greatest element of the left side to make it connected. The long line can also refer to the right side of the long line, with the least element removed to make it unbounded. I'll use "long line" to refer to the last one from now on.
If α is an infinite countable ordinal, then (1+θ)α = 1+θ. If α > ω₁, then (1+θ)α with the smallest point removed is no longer a homogeneous linear order (I call an LO α homogeneous iff ∀x,y ∈ α ∃f ∈ Aut(α) f(x) = y). ω₁ is thus purrfect for making a long line.
Here is a funni connected order I came up with: (1+θ+1)θ.
Here are some theorems that state more generally how arithmetic works with connected linear orders:
α+β is connected iff α is connected, β is connected, and either α is right-bounded and β is left-unbounded, or α is right-unbounded and β is left-bounded;
α* is connected iff α is connnected;
For unbounded α, αβ is connected iff α is connected and [β = 0 or β = 1];
For bounded α, αβ is connected iff α and β are connected;
For left-bounded right-unbounded α, αβ is connected iff α is connected, every S ⊂ β with an upper bound has a supremum (least upper bound), and every point x ∈ β has a direct next point y ∈ β (∄z ∈ β x < z < y).
I like the last one :3 You can try proving these theorems if you want to.
Bye!~
I hope I've given you good intuition on the most common linear order types ω, ζ, η and θ ^^ If you spot any mistakes in my post, please tell me!
I'm planning to write an introduction to set theory next :3