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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.
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    *Complemented* is the word you want, unless the elements of the lattice are very very polite. – 2017-02-04

1 Answers 1

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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.

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    yeah I know first point is kinda stupid, it was direct implication of the quoted fact. In fact I came across following (confusing/possibly wrong) statement in one of the sources: a "complemented" lattice is distributive iff every element has a unique complement. Its not even like that right?. (Noob in lattice theory... :\) – 2017-02-05
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    The "Fact" you quote is false in general, but it is true for finite lattices. – 2017-02-05