False: every ordered set has a minimum.
False: every ordered set has a maximum.

False: every ordered set has a maximal element.


But:

True: every finite ordered set has a maximal element.

************ PROVE IT!!!!!!!!!!!!!!!!!! *****************

"take any x; if x is not maximal, replace x by some y > x; repeat until done"


x := any element
while (there is some y > x):
   x:= y
end while
return x

Since the set is finite, you will terminate eventually.

Termination function f(x) that (1) is always >= 0 and (2) decreases in every step
For example, f(x) := "the number of y that are > x"

Observation: If x < y than f(y) <= f(x)-1

because f(x) counts everything that f(y) counts, but it counts more, for example, it counts y.





                   Infinitely larger versus arbitrarily large





                   What is the largest antichain?


v                  Can you find an antichain of size 1000?


                   Look at A_k := { (x,y) | x+y = k }

                   This has size k+1, so they become larger and larger.
                   However, every A_k is finite.


                   { A_1, A_2, A_3, ... }
                 
                   this is a set of sets
                   it is an infinite set
 
                   every element of it is a finite set
                   
                   of all its elements, which set is largest?
                   this question makes no sense: it contains arbitrarily large sets.





                   R = (N x N,   {(a,b) | a divides b} )

                   True or false: R contains an infinite chain.
                   True: {1,2,4,8,16,...} = {2^i | i >= 0} = the powers of two

                   True or false: R contains an infinite antichain
                   Yes, all the prime numbers.

                   



                   X = {  (x,y) in NxN | x >= y}

                   R = (X, { ((a,b),(u,v)) in XxX | a=u, b<=v} )


                   Does it have an infinite antichain?
                   
                   Yes: { (1,1), (2,2), (3,3), ...}

                   Does it have an infinite chain?

                   Arbitrary, but not infinite

                   (1) arbitrary: of size k?    (k,1), (k,2), ..., (k,k)

                   (2) infinite: no. Take an element (u,v) in this chain.
                   The chain has at most u elements: 
                   it must be a subset of (u,1), (u,2), ..., (u,u).


                   3.3: Does NxN have an infinite antichain?
                    
                   It seems not. But how to prove?


                   Latex: $x \preceq y$




    Hasse diagram:

   If R is a partial order, then the Hasse diagram R! is defined as

   R! := { (x,y) | x < y and there is no z in between, i.e., no z with x < z < y}


   "The minimum information you have to give me such that I can reconstruct R"

   Not completely true. What's the Hasse diagram of "<=" on the real numbers? It's EMPTYYYYY!



   I guess the following is true, but I have not bothered to think about it a lot:

   THEOREM: If R is a finite ordering, then (R!)^* = R.
   Furthermore, (R1)^* = the intersection of all relations S with S^* = R.
   This is math for "It is the smallest relation from which you can recover R"