1,382 views
1 votes
1 votes
If X and Y are distributive lattice then show that X×Y are also distributive lattice ?

2 Answers

2 votes
2 votes

Suppose X is distributive lattice having $x_{1},x_{2},x_{3}........x_{n}$ elements and

Suppose Y is distributive lattice having $y_{1},y_{2},y_{3}........y_{m}$ elements

then total number of elements in X $\times$ Y =  n$\times$m

A non distributive lattice does not satisfy the distributive property because it contains 2 or more pair of elements having common LUB and GLB. 

So if X and Y are distributive lattice 

then if $x_{i}$ and $x_{j}$ have GLB $x_{k}$ and LUB $x_{l}$ and

if $y_{i}$ and $y_{j}$ have GLB $y_{k}$ and LUB $y_{l}$

then in Lattice X $\times$ Y,

elements $\left ( x_{k}, y_{k} \right )$ and $\left ( x_{l}, y_{l} \right )$ can be GLB and LUB of only 2 elements $\left ( x_{i}, y_{i} \right )$ and $\left ( x_{j}, y_{j} \right )$

this applies on every pair of Lattice X $\times$ Y. Hence Lattice X $\times$ Y holds distributive property and hence is a Distributive Lattice.

reshown by
0 votes
0 votes

if X = {1,2,3,6 ; /} and  Y = {1,2,4 ; /}..Since both X and  Y are distributive then how to represent X×Y because it contains ordered pairs after cartesian product.

 here we can take Cartesian product of two boolean algebra first

X*Y={1,2,3,4,6,8,12,24;/} because both set have common elements number of elements we will get is 8 only not 12 as expected 

reshown by

Related questions

0 votes
0 votes
0 answers
1
Nidhi Budhraja asked Aug 22, 2018
797 views
Boolean algebra is a bounded distributed complemented lattice, also lattice is a poset thus it satisfies reflexive , antisymmetric and transitive properties. Does Boolean...