lattice

Question

i know that In distributive lattice , a given element can have at most one complement.

 

what if there are some elements in lattice for which complement doesn't exist?
what i mean to ask is, here AT MOST means element can have <= 1 complement , so there may some elements which has 0 ( means < 1 ) complement. then also that lattice is called as distributive?

Comments

i just want to know that for a lattice to be distributive does it neccessary that every element must have at most one complement?

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yes it is necessary condition but not the sufficient condition .

you should also check the distributive property

Answer 1

In a distributed lattice, each element has 0 or 1 complement. It is not necessary that each element should have a complement.

In a complemented lattice, all elements have 1 or more complements.

A complemented distributive lattice is called a Boolean Algebra. In it, all elements have exactly one complement.