Question

Show that any complete lattice has a zero and an element ​

Answer 1

To prove that every chain ⟨P,⩽⟩⟨P,⩽⟩ is a lattice, fix some a,b∈Pa,b∈P and w.l.o.g assume that a⩽ba⩽b. From reflexivity of ⩽⩽ it follows that a⩽aa⩽a, hence aa is a lower bound of the set {a,b}{a,b}. To prove that it is the greatest lower bound note that if some c∈Pc∈P is another lower bound of {a,b}{a,b} then by the definition of a lower bound we have c⩽ac⩽a. It means that aa is the greatest lower bound of {a,b}{a,b}. Same reasoning shows that bb is the least upper bound of {a,b}{a,b}.

To prove that every chain ⟨P,⩽⟩⟨P,⩽⟩ is distributive, you should just consider all possible relations between three arbitrary elements a,b,c∈Pa,b,c∈P and check that distributive identity holds.

For example, let a⩽b⩽ca⩽b⩽c, hence a∧b=a∧c=aa∧b=a∧c=a and b∨c=cb∨c=c, so

a∧(b∨c)=a∧c=a=a∨a=(a∧b)∨(a∧c).