Am I Understanding This Correctly Partial Order Vs Total Order Sets

Can all large cardinals be arranged in a total order or do they have contradicting properties (hence constituting a partial order at best)?

The question seems to be about large cardinal axioms, which is quite different from the question of whether any set can be well-ordered (as other answers seem to assume.)As far as we know, all large cardinal axioms sort themselves by consistency strength into a linear order, but it is considered something of a mystery as to why that is.There are a few instances where we aren’t quite sure what is stronger than what, but as far as I know there are no known cases of natural LC axioms which are mutually incompatible.On the other hand, there’s no obvious reason why this can’t happen. There’s also no precise definition of what a LC axiom is allowed to be (though Woodin has made important suggestions[1] in this direction.) This makes it hard to even formalize the problem.It is surprising and intriguing that LC axioms tend to arrange themselves in such a linear order. It creates a strong sense that there’s an underlying phenomenon we are failing to observe.Footnotes[1] Ω-logic - Wikipedia

How do I understand partially ordered sets?

I think the question is very poorly worded, so I cannot be certain I am answering this correctly. That said, I will show you the ideas, and then let you deal with the actual question.My Assumptions:[math]\Sigma[/math] is the set of all finite length strings written in the alphabet [math]\{0,1\}[/math],The questioner wants you to use the dictionary order on strings, where we take letter [math]0[/math] to be less than letter [math]1[/math]. Note that the question implies you can compare strings of different lengths, but the questioner does not say how, SO, the next point as well.For calculations, you should actually pad out any two words you wish to compare by adding and invisible letter ([math]-1[/math]) to the end of the shorter word in comparison, so that you always compare words of the same length. By assumption, [math]-1<0<1[/math] as letters for the comparison. This will allow you to see that the word 101 is less than 1010, for instance (as happens in a dictionary).Now, our rules allow us to see that 01101 < 011101 since the fourth letters determine this is the case. Therefore, any down set containing 011101 contains 01101 for free. Therefore, the smallest downset containing both is actually the downset with (unique!) maximal element 011101. That is, we want the set of all strings [math]w[/math] that have [math]w\leq 011101[/math].

Is there a natural way to put a total order on the set of all algebraic number fields?

Not quite (not in any way I can see), but you can come close.You can first sort number fields [math]K[/math] by their degree [math]|K:\Q|[/math]. The next natural invariant is the discriminant, which can reasonably be interpreted as a measure of “size” which is more refined than the degree.Unfortunately, the discriminant isn’t sufficient to distinguish all number fields (though it is sufficient for quadratic ones). But there is good news: there are only finitely many fields with a given degree and discriminant [1] . This follows from a theorem[2]of Hermite and Minkowski. Therefore, sorting number fields by degree and discriminant “almost” totally orders them: the batches of incomparable fields are finite.We may continue to add more invariants, such as the regulator[3], to refine that order. However, there are pairs of non-isomorphic number fields which are uncannily similar: they have the same zeta function, and in particular the same degree, discriminant and regulator. I’m not sure any (reasonable) set of invariants suffices to tell all pair of fields apart.Footnotes[1] Discriminant of an algebraic number field - Wikipedia[2] Hermite–Minkowski theorem - Wikipedia[3] Regulator of an algebraic number field

When describing the ZooKeeper atomic broadcast, why are both "Total Order" and "Causal Order" mentioned?

I think I asked the question too soon. It was right in front of me.The first mistake I did was that I confused Causal ordering with Partially ordered set. The concept is similar but they are not the same.There is a beautiful paper on ZAB, "A simple totally ordered broadcast protocol. Pasting relevant snippet here:Reliable delivery: If a message, m, is delivered by one server, then it will be eventually delivered by all correct servers.Total order: If message a is delivered before message b by one server, then every server that delivers a and b delivers a before b.Causal order: If message a causally precedes message b and both messages are delivered, then a must be ordered before b.….1. The reliability and total order guarantees ensure that all of the replicas have a consistent state;2. The causal order ensures that the replicas have state correct from the perspective of the application using Zab;So what it essentially means Zab needs Total order to ensure atomic broadcast. Causal order is needed for efficient working of Zookeeper.Causal order means if message a cause message b, then there is no way b should occur before a. This concept is necessary in Zookeer for leader's write request.References:If this is still confusing, then along with paper mentioned above following two are very good articles which will help to clear your understanding:A nice explanation on causal ordering: Scattered ThoughtsA nice explanation of causal ordering in context of Zookeeper:  { work }: Causal Atomic Broadcast

Can we define a sorting or an ordering for a set A under a relation R which is not transitive but is anti-symmetric and reflexive?

A relation which is anti symmetric and reflexive but not transitive seems like a very obscure relation. Although, there exist some not so elegant examples[1] of such a relation, its not clear to me why would you think about sorting a set where there is not even a hint of a (partial) order.There is nothing that stops you from implementing such a relation, but you can’t really predict the behaviour of any sorting algorithm on it. For instance STL’s sort[2] assumes that underlying list or set has a weak ordering[3], and sorting it with a comparison function that meets your demands can result in non deterministic runs of the algorithm.Footnotes[1] non-transitive, antisymmetric and reflexive binary relation on $\mathbb Z$[2] sort - C++ Reference[3] Weak ordering - Wikipedia

How does one describe the partial order relation that puts the words in a dictionary in the right order?

That’s the lexicographical order. Note that the standard lexicographical order is a total order, and requires that the underlying set of “letters” be totally ordered.