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Orders
Now let's talk about order. Given a set of objects, there can be numerous criteria, based on which to order them, and which depend on the objects themselves - size, weight, age, alphabetical order etc.
Howevever, as mathematicians we are not interested in the criteria that we can use to order objects, but in the nature of the relationship that defines the order, of which there can be several types as well.
Mathematically we can represent order as a set of things (e.g. our balls) and a binary relation between these things (which we often represent as a bunch of arrows). Not all binary relationships are orders - only ones that fit certain criteria that we are going to examine as we review the different types of order.
Linear order
The most straightforward type of order that you think about is linear order, one in which every object has its place depending on every other object, i.e. the ordering criteria is completely deterministic and leaves no room for ambiguity in terms of which elements comes before which. For example, ordering the colors by the length of their waves (or by how they appear in the rainbow).
In most programming languages, we can order objects lineary by providing a function which, given two objects, outputs us which one of them is "bigger" (comes first) than the other.
[1, 3, 2].sort((a, b) => {
if (a > b) {
return true
} else {
return false
}
})
But in order for such function to really define an order (e.g. give the same output every time, independent of how were the objects shuffled initially), it has to obey several rules. Incidentally, (or rather not incidentally at all), these rules are nearly equivalent to the mathematical laws that define the criteria of the relatioship between elements in an order i.e. those are the rules that define which element can point to which. So let's review them.
Reflexivity
Let's get the most boring law out of the way - each object has to be bigger or equal to itself, or a ≤ a (the relationship between elements in an order is commonly denoted as ≤ in formulas, but it can also be represented with a simple arrow from first object to the second).
No special reason for this law to be so, except that the "base case" should be covered somehow.
We can formulate it the opposite way too and say that each object should not have the relationship to itself, in which case we would have a relation than resembles bigger than, as opposed to bigger or equal to and a slightly different type of order, sometimes called a strict order.
Transitivity
The second law is maybe the least obvious, (but probably the most essential) - it states that if one object one bigger than object two, it is automatically bigger than all objects that are smaller than object two or a ≤ b and b ≤ c ➞ a ≤ c.
This is the law that to a large extend defines what an order is: if I am better at playing soccer than my grandmother, then I would also be better at it than my grandmother's friend, whom she beats, otherwise I wouldn't really be better at her.
Antisymmetry
The third law is called antisymmetry and it states that the function that defines the order should not give contradictory results (a ≤ b ⟺ b ≰ a).
It also means that no ties are permitted - either I am better than my grandmother at soccer or she is better at it than me.
Totality
The last law is called totality (or connexity) and it mandates that all elements that belong to the order should be comparable - a ≤ b or b ≤ a. That is, for any two elements, one would always be "bigger" than the other.
By the way, this law makes the reflexivity law redundant, as it is just a special case of reflexivity when a and b are one and the same object. Why do we introduce reflexivity then? We shall see soon.
You might say that this law is not as self-evident as the rest of them - if you think about different types of real-life objects that we typically order by various criteria, you would probably think of some situations in which it does not apply. For example, if we aim to order all people based on soccer skills there are many ways in which we can rank a person compared to their friends their friend's friends etc. but there isn't a way to order groups of people who never played with one another.
Orders that resemble the order of people based on their soccer skills in that they don't follow the totality law are called partial orders and they are actually much more interesting than linear orders (which are also called total orders, by the way).
Task: Think about some orders that you know about and figure out whether they are partial or total.
But before we dive into the world of partial orders, let's say a few things about the numbers.
The order of natural numbers
Natural numbers form a linear order under the operation bigger or equal to (the symbol of which we have been using in our formulas).
In many ways, numbers are the quintesential order - every finite order of objects is isomorphic to a subset of this order, as we can map the first element of any order to the number 1, the second one to the number 2 etc (and we can do the opposite operation as well) and if we think about it, this isomorphism is actually closer to the everyday notion of order, than the proper definition (the one defined by the laws) - when most people think of order, they aren't thinking of a transitive, antisymmetric and total relationship, but are rather thinking about criteria based on which they can decide which object comes first, which comes second etc.
From the fact that any finite order of objects is isomorphic to the natural numbers it also follows that all linear orders of the same magnitude are isomorphic to one another.
Question: If all orders are isomorphic with one another, why do we say that numbers are the quintessential order?
So the linear order is a perfect order, but it is also (and I think that these theorems prove it) the most boring order ever, especially when looked from a category-theoretic viewpoint - all finite linear orders are just isomorphic to the natural numbers (by the way, most infinite orders are isomorphic to the natural numbers as well, with the exception of the ones for which Cantor's diagonal argument applies) and so all of their diagrams look in the same way.
However, this is not the case with partial orders that we will look into next.
Partial order
Like linear orders, a partial orders consists of a set plus a relationship, with the only difference that, although it still obeys the reflexive, transitive and the antisymmetric laws, the relationship does not obey the law of totality - not all elements are necessarily ordered. I say "necessarily" because even if all elements are ordered, the partial order is still a partial order (just as a group is still a monoid) - all linear orders are also partial orders, but not the other way around (we can create an order of orders, based on which is more general, but that is probably too meta).
So let's revisit the example of the soccer players ranglist. The first version that includes just myself, my grandmother and her friend is a linear order.
However, including this other person whom none of us played yet, makes the hierarchy non-linear.
Before, we said that all linear orders can be represented by the same chain-like diagram, we can reverse this statement and say that all diagrams that look something different than the said diagram represent partial orders (or preorders).
Let's see some examples.
Chains
A partial order can contain diffranked erent linearly-ordered subsets, e.g. in our soccer example, we can have separate groups of friends who play together and are ranked with each other, but not with anyone from other groups.
The different linear orders that make up the partial order are called chains. There are two chains in this diagram M ➞ G ➞ F and D ➞ O.
The chains in an order don't have to be completely disconnected from each other in order for it to be partial, we can have connections as long as they are not all one-to-one i.e. when the last element from one chain should not be connected to the first element of the other one (as this would effectively unite them into one chain). But there are other types of connections, like one-to-many and many-to-one.
This set is not lineary-ordered - although the connection establishes the relationship between D and G (D ≤ G) and although the relationship between F and G is known as well (F ≤ G), the relationship between D and F is not known. Any element can be bigger than the other one.
Maximum and minimum
As we saw, partial orders cannot provide us with a definite answer of the question who is better at who. But that is sometimes OK, as in sports, as well as in other domains, people often don't care about who is better than who - they only care who is number one, who is the champion, the player who is better than anyone else, or more generally the element that is bigger than any other element.
We call such elements the maximum element and some partial orders do have such element - in our last diagram M is the maximum element, in this diagram, the green element is the biggest one.
Sometimes we have more than one elements that are bigger than all other elements, in this case none of them are maximum.
In addition to the maximum element, a partial order may also have a minimum (smallest) element, which is defined in the same way.
Joins
The least upper bound of two elements that are connected as part of an order is called the join of these elements, e.g. the green element is a join of the other two.
There can be multiple elements bigger than A and B (all elements that are bigger than C are also bigger than A and B), but only one of them is a join. Formally, the join of A and B is defined as the smallest element C that is bigger than both A and B (i.e.for which A ≤ C, and B ≤ C.
Given any two elements in which one is bigger than the other (e.g. A ≤ B), the join is the bigger element (in this case B).
In a totally ordered set, the join of any subset of elements is just their the maximum element.
Like with the maximum element, if two elements have several upper bounds that are equally big, then none of them is a join (a join must be unique).
If, however, one of those elements is established as bigger than another, it immediately qualifies.
Question: Which concept in category theory reminds you of joins?
Meets
Given two elements, the biggest element that is smaller than both of them is called the meet of these elements.
The same rules as for the joins apply.
Interlude: Hasse diagrams
In this section, we use what are called "Hasse diagrams" - they are much like our usual diagrams, however they have an additional rule that is followed - "bigger" elements are always positioned above smaller ones.
In terms of arrows, the rule means that if you add an arrow to a point, the point to which the arrow points must always be above the one from which it points.
This arrangement allows us to compare any two points by just seeing which one is above the other e.g. to spot the join of two elements, you just have to identify the ones they connect to and see which one is lowest.
Examples of partial orders
We all know many examples of total orders (any form of chart or ranking is a total order), but there are probably not so many obvious examples of partial orders that we can think of off the top of our head. So let's cook some. In addition to providing a little context, this will help us understand joins and see why are they significant.
Color order
To stay true to our form, let's revisit our color-mixing monoid and create a color-mixing partial order in which all colors point to colors that contain them.
If you go through it, you will notice that the join of any two colors is the color that they make up. Nice, right?
Numbers by division
When we order numbers by "bigger or equal to", they form a total order (the total order even). But numbers can also form a partial order, for example if we order them by the relationship of which divides which, like if A divides B, then A is before B e.g. because 2 * 5 = 10 2 and 5 come before 10 (but 3, for example does not).
An it so happens (actually for very good reason) that the join operation again corresponds to an operation that is rellevant in the context of the objects - the join of two numbers in this partial order is their least common multiple.
Meanwhile, the meet (the opposite of join) of two numbers is their greatest common divisor.
Inclusion order
Given a collection of all possible sets containing a combination of a given set of elements (like for example our colourful balls)...
We can define what is called the inclusion order of those sets, in which A comes before B if A includes B, or (to use the term from) chapter 1, if B is a subset of A.
Note that the join operation in an inclusion order is the set union, and the meet operation as set intersection.
This diagram might remind you of something, because if we take the colors that are in each of its set and mix it into one color, we get the color-blending poset that we saw earlier.
The poset example with the the number dividers is also isomorphic to an inclusion order. It is the inclusion order of all possible sets of prime numbers, including repeating ones. This is confirmed by the fundamental theory of arithmetic, which states that every number can be written as a product of primes in exactly one way.
And, to be more precise, the isomorphism between the number poset and the prime inclusion order is comprised of (as any other isomorphism) one function from the prime inclusion order, to the number poset (which in this case is just the multiplication of all the elements in the set) and one function from the number poset to the prime inclusion order (which is an operation called prime factorization of a number, consisting (unsurprisingly) of finding the set of prime numbers that result in that number when multiplied with one another (as we said, this set is unique, and hence these functions realy do comprise an isomorphism)).
Birkoff's representation theorem
So far we saw two partial orders of different elements (one based on color mixing, and one based on number division) which can be represented by the inclusion orders of all possible combinations of sets of more basic elements (the primary colors in the first case, and the prime numbers (or prime powers) in the second one). Many other partial orders can be defined in this way. Which ones exactly is a question that is answered by an amazing result called Birkoff's representation theorem in which the author proved that they are the partial orders that meet the following two criteria:
- All elements have joins and meets (those orders are called lattices, by the way)
- Those two operations are distribute over one another (we will see what that means shortly)
I won't go into details about this, I would only mention that the "prime" elements with which we can construct the inclusion order are the ones that are not the join of any other elements.
By the way, the partial orders that are NOT distributive lattices are also isomorphic to inclusion orders, it is just that they are isomorphic to inclusion orders that do not contain all possible combinations of elements.
Preorder
As interesting partial orders are, I like to leave them aside for a sec just so I can return back to them, like in an old romance novel.
First off , I like us to cover one more kind of order (or something like it). In the previous section, we saw how removing the law of totality from the laws of (linear) order produces a different (and somewhat more interesting) structure, called partial order. Now let's see what will happen if we remove another one of the laws, namely the antisymmetry law. If you recall, the antisymmetry law mandated that you cannot have an object that is at the same time smaller and bigger than another one. (or that a ≤ b ⟺ b ≰ a).
| Total order | reflexive | transitive | antisymmetric | total | Either a ≤ b or b ≤ a
| Partial order | reflexive | transitive | antisymmetric | total | Either a ≤ b or b ≤ a or they are unrelated.
| Preorder | reflexive | transitive | antisymmetric | total | Both a ≤ b and b ≤ a are possible.
This results in something called a preorder which is not exactly an order, as I hinted at the beginning - it can have arrows coming from any point to any other: if a partial order can be used to model who is better at who in soccer, then a preorder can be used to model who has beaten who, either directly (by playing him) or indirectly.
Preorders have just one law - transitivity a ≤ b and b ≤ c ➞ a ≤ c (two, if we count reflexivity). The part about the indirect wins is a result of this law. Due to it, all indirect wins (ones that are wins not against the player directly, but against someone who had beat them) are added as a direct result of its application, as seen here (we show indirect wins in lighter tone).
And as a result of that, all "circle" relationships (e.g. where you have a weaker player beating a stronger one) result in just a bunch of objects that are all connected to one another. Using this effect, we can convert the preorder into partial order by grouping all objects that have arrows to one another in sets and then create an order from those sets (they are called equivalence classes, by the way).
All of that structure arizes naturally from the simple law of transitivity.
Orders as categories
Now let's look at transitivity law again, but from a different perspective. What it tells us that if we have two pairs of relationship a ≤ b and b ≤ c, then we automatically have a third one a ≤ c. In other words, it tells us that the ≤ relationship composes i.e. if we view the "bigger than" relationship as a morphism we would see that it fits the categorical definition of composition.
(we have to also verify that the relation is associative, but that's easy)
What about that other law that was required in order to be a category - the identity law? We have it too, under the name reflexivity.
So it's official - preoders are categories (sounds kinda obvious, especially after we also saw that orders can be reduced to sets and functions (the inclusion order) and sets and functions form a category at their own right). And since partial orders and total orders are preorders too (as they obey those two laws), they are categories as well.
Starting to compare the categories of preorders, and partial and linear orders to other categories, like the (quintessential) category of sets, immediately sets them apart. In other categories there can be many different morphisms (arrows) between two objects and in orders can have at most one morphism. That is, for two objects a b we either we have a ≤ b or we do not.
That is in the contrast with the category of sets where there are potentially inifinite amount of functions from, say, the set of integers and the set of boolean values, as well as a lot of functions that go the other way around, and the existence of either of these functions does not imply that one set is "bigger" than the other one.
Products and sums
While we are rehashing diagrams from the previous chapters, let's look at the diagram defining the sum of two objects in a category.
If you recall, this is an operation that corresponds to set inclusion in the category of sets.
But wait, wasn't there something else that corresponded to set inclusion - oh yes, the join operation in orders. And not merely that, but orders are defined in the exact same way as the categorical coproducts.
In category theory, an object G is the coproduct of objects Y and B if the following two conditions are met:
- We have a morphism from any of the elements of the coproduct to the coproduct, so Y → G and B → G.
- For any other object P that also has those morphisms, so for any P such that P ≤ G and P ≤ B, we would have morphism G → P.
In the realm of orders, we say that G is the join of objects Y and B if:
-
It is bigger than both of these objects, so Y ≤ G and P ≤ G.
-
It is smaller than any other object that is bigger than them, so for any other object P* such that P ≤ G and P ≤ B then we should also have G ≤ P.
We can see that the two definitions and the diagrams are the same, so in category theory language, we can say that the categorical coproduct in the category of orders is the join operation.
Logic
So here we go with this little leit motif of mine where I begin talking about something completely different and then it turns out that it is all connected. This time I will not merely transport you to a different branch of mathematics, I will talk about an entirely different discipline, namely logic. Or, to be more precise, intuitionistic logic. This discipline may seem to you as detached from what we have been talking about as it possibly can, but it is actually as close as it can - I don't even have to start another chapter for it (althought it will make this one a bit lenghty).
Logic aims to it study of the rules by which knowing one thing leads you to conclude that some other thing is true, and most importantly, to do that in the formal way i.e. without considering what these things are.
Not only that, but logicians try to organize those rules in what are called systems, of formal systems - containing selections of rules and also of rules for applying those rules (rules of inference) selected so they have the expressive ability to prove everything that you can see by intuition.
What does "prove" mean in this context? Simple, when we are able, using the rules of a given logical system to transform one set of assertions A to another one B we say that we have proven that A → B in that logical system.
And
So let's say I know a given fact, does not matter what it is or how exactly I learned it,
and say that it is an "atomic fact" as Wittgenstein calls it - it is not composed of any "more basic" facts, it is just the simplest thing that you can know.
So let's get our balls again.
Or and And relations
False