POSET, HASSE diagrams and Lattices

 when there is partial ordering, we represent partial ordering by hasse diagrams

Example of partial ordering: Dressing Up

we cannot wear a tie before wearing a shirt. There is partial ordering between wearing a tie and wearing a shirt

                                   shirt

                                      |

                                    tie

we cannot wear shoes before wearing pants.          

                                   pants

                                      |

                                    shoes

Similarly

                                   pant

                                      |

                                    belt

but there is no ordering required to  between wearing left shoe and right shoe. We can also assume there is no ordering required in wearing shirt and wearing pants. Hence our complete hasse diagram to represent partial ordering in dressing up is shown below

To check whether a given hasse diagram is lattice or not:

1. every pair must have unique least upper bound (luh or supremum or join ) and greatest lower bound (glb or meet or infimum)

for example:

upper bound of pair belt,tie ={belt, shirt, pant}

i.e, all that are above tie

least upper bound = belt

(immediately above tie)

lower bound of pair belt, tie={tie,left sock, right sock, left shoe, right shoe}

greatest lower bound of pair belt,tie = tie

but the hasse diagram is not lattice as for the pair {left sock, right sock} there is no lower bound

One real life example of lattice is when we consider natural numbers partially ordered by divisibility, the unique supremum is LCM of those two numbers and unique infimum is GCD. (as we know that GCD and LCM is unique for a given pair of numbers

https://www.slideshare.net/rupalirana07/ch-2-lattice-boolean-algebra