POR: Partial ordered relation
GLB: Greatest lower bound
LUB: least upper bound
Let’s analyze the options
B) False
It is a Lattice ,since for every pair of elements GLB and LUB exist.
A) True
Since neither pentagon nor kite structure will never be possible in sublattice of given lattice , as it is total order lattice. Thus given Lattice is Distributive lattice .
For a lattice to be called complemented lattice it should satisfy two main properties ,
It should be bounded i.e. It should have least element and greatest element.
GLB and LUB performed on any pair of element of the given set should always give respective least element and the greatest element , simultaneously.
Since given relation defined on a set is Total ordered relation. In addition , it is not bounded. We know that in total ordered reation (given in Question) greatest element doesn’t seems to be fixed. Thus it is not complemented Lattice.
C) True
For a Lattice to be called boolean lattice it should have properties such as bounded,complemented and distributive.
Given POR is not bounded ,hence no question of boolean lattice.
D) True
For an element to have complement , its GLB and LUB with the paired element should fetch least element and greatest Element respectively .
Greatest element is not fixed i.e. notbounded/infinite .
Thus Element O doesn’t have complement
Finally we are asked about False statement . Hence Option B is correct option.
