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Consider the lattice D30

a)Draw the Hasse Diagram of D30.

b) Is D30 complemented?

c) Is D30 distributive ?

asked in Set Theory & Algebra by Boss (5.7k points)  
recategorized by | 226 views

2 Answers

+6 votes
Best answer

(a)

 

(b)In a Complemented Lattice each element has at least one complemented.

(1) 1∨30 =30                  1∧30 = 1         $\Rightarrow$ 1' =30 and 30'=1

(2) 215 = 30                 2∧15 = 1         $\Rightarrow$  2'=15 and 15'=2

(3)310 = 30                  3∧10 = 1         $\Rightarrow$ 3'=10 and 10'=3

(4)56 = 30                    5∧6 = 1            $\Rightarrow$ 5'=6 and 6'=5

Here Every element has one element hence,it is a complemented Lattice.

 

(c)In a Distributive Lattice complement of an element if exists is unique i.e. each element has at most one element.

Here in D30 Every element has unique  complement.Hence, it is Distributive Lattice.  

 

answered by Veteran (31.3k points)  
selected by
very good explanation..
0 votes

D30={1,2,3,5,6,10,15,30}

because D30 is square free so it is boolean algebra and every  boolean algebra is distributed and complemented also

answered by Boss (9.4k points)  
what do you mean by square free ?


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