Unique complements in a distributive lattice

Question

I know that

In a distributive lattce L, a given element can have at most one complement.

But are following points correct:

• In complimented distributive lattices, complements are unique.
• If compliments are unique in a lattice then it must be a distributive lattice.

Answer 1

Your bullet point 1 is correct, and it follows from your statement:

In a distributive lattice $L$, a given element can have at most one complement.

[If $L$ is a complemented lattice, each element has at least one complement. As you note, in the distributive case it has at most one complement. Together these mean the lattice is uniquely complemented.]

Your bullet point 2 is not correct. It is a famous result of Dilworth that every lattice is embeddable in a uniquely complemented lattice. See

R.P. Dilworth, Lattices with unique complements, Trans. Amer. Math. Soc. 57 (1945), 123-154.