Lattice | Distributive

Question

Identifying a Distributive Lattice

If it is isomorphic to two standard lattice L1* and  L2*  then the lattice is Not Distributive .
If Complemented Then unique complement should exists .

Identifying a complemented Lattice

Atleast one complement exists for every pair of element . (a,b) are complement to each other if meet and loin of (a,b) are Upper Bound and Lower Bound of lattice .

Is My understanding Correct ??

Q1 ]  POSET [{ 1,2,3,6,12}; / ]
         This POSET is a distributive lattice Eventough complement does not exists for 2,3 .
        IF Complement  exists it should be UNIQE. Atmost one complement for every element
       Is my analysis correct ?
Q2]  POSET [{1,2,3}, <= ]

      What is complemet of 1 , 2 and 3 ? Im not able to find out with above understanding
     

Answer 1

1. 1 ----> (2, 3) ---> 6 ----> 12, here complement of 12 is 1. For all other elements complement doesn't exist.

2. 1 ---> 2 ---> 3, here complement of 3 is 1 and complement of 2 doesn't exist.

Comments

Thank you sir

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basically if a lattice contains L1* or L2*, it is not distributive. Right ??

Answer 2

IN Distributive Lattice atmost one complement of each element should be present.

But In complemented lattice at least one complement of each element should be present.