Question

If lattice (C,<=) is a complemented chain, then
a) |C|<=1
b) |C|<=2
c) |C|>1
d) doesn't exist.

Answer 1

Every chain is bounded: 0 and 1 belongs to C, or they coicide, so |C|=1.
If there is third element a, then its complement is such b, that inf(a,b)=0 and
sup(a,b)=1. But as it is chain, then we have a<=b or b<=a. If a<=b then
0=inf(a,b)=a and 1=sup(a,b)=b. Case b<=a is similar.

So there cannot be other third element in C.

|C|=2 or |C|=1 => |C|<=2