Lattice

  Hasse Diagram:

                    Hasse diagram is used to represent POSET in a diagrammatic way! For each subset of Hasse Diagram there may be one or more upper bounds and lower bounds.

For example consider the following Hasse diagram. For example the subset {a,b} have upper bounds b,d,e,f and lower bounds a.
The subset {b,c} have upper bounds d,e,f,g and lower bounds a.

Lattice:
                    
                             For every subset of POSET there exists many lower and upper bounds. A lattice is defined on POSET as for every subset of Hasse Diagram there must be one least upper bound and greatest lower bound.

For example consider the following diagram

The set {a,b} have least upper bound 'b' and greatest lower bound 'a'

The set { b,c} have least upper bound 'd' and greatest lower bound  'a'

The set {e,f} have least upper bound 'g' and greatest lower bound 'd'

The set {e,d} have least upper bound 'e' and greatest lower bound 'd'

Similarly the remaining subsets can be proved

Hence the given POSET is Lattice


Consider another Hasse Diagram given below.

The above diagram is not Lattice since

for example the set {b,c} does not have least upper bound.(since we cannot compare e and f)



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