Question

Prove that every chain is a distributive lattice

Answer 1

From reflexivity of it follows that a ⩽ a , hence is a lower bound of the set . ... Same reasoning shows that is the least upper bound of . To prove that every chain is distributive, you should just consider all possible relations between three arbitrary elements a , b , c ∈ P and check that distributive identity holds.

Answer 2

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

I HOPE IT HELPFUL FOR YOU.
PLEASE MAKE MY ANSWER BRAINLIEST.