<< Chapter < Page Chapter >> Page >

The poset of the set of positive real numbers with the less-than-or-equal-to relation is not a well order, because the set itself does not have any least element (0 is not in the set).

A digraph of a binary relation on a set can be simplified if the relation is a partial order. Hasse diagrams defined as follows are such graphs.

Definition(Hasse diagram): A Hasse diagram is a graph for a poset which does not have loops and arcs implied by the transitivity. Further, it is drawn so that all arcs point upward eliminating arrowheads.

To obtain the Hassse diagram of a poset, first remove the loops, then remove arcs<a, b>if and only if there is an element c that<a, c>and<c, b>exist in the given relation.

Example 10: For the relation {<a, a>,<a, b>,<a, c>,<b, b>,<b, c>,<c, c>} on set {a, b,c}, the Hasse diagram has the arcs {<a, b>,<b, c>} as shown in Figure 10.

Topological sorting

The elements in a finite poset can be ordered linearly in a number of ways while preserving the partial order. For example {∅, {1}, {2}, {1, 2}} with the partial order ⊆, can be ordered linearly as ∅, {1}, {2}, {1, 2}, or ∅, {2}, {1}, {1, 2}. In these orders a set appears before (to the left of) another set if it is a subset of the other. In real life, tasks for manufacturing goods in general can be partially ordered based on the prerequisite relation, that is certain tasks must be completed before certain other tasks can be started. For example the arms of a chair must be carved before the chair is assembled. Scheduling those tasks is essentially the same as arranging them with a linear order (ignoring here some possible concurrent processing for simplicity's sake).

The topological sorting is a procedure to find from a partial order on a finite set a linear order that does not violate the partial order. It is based on the fact that a finite poset has at least one minimal element. The basic idea of the topological sorting is to first remove a minimal element from the given poset, and then repeat that for the resulting set until no more elements are left. The order of removal of the minimal elements gives a linear order. The following algorithm formally describes the topological sorting.

Algorithm Topological Sort

Input: A finite poset<A, R>.

Output: A sequence of the elements of A preserving the order R.

integer i;

i := 1;

while ( A ≠∅) {

    pick a minimal element b from A;

    A := A - {b};

    i := i + 1;

    output b

} Example: Let A = {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} with the partial order ⊆. This given A has three minimal elements {1}, {2}, and {3}.

Select {2} and remove it from A. Let A denote the resultant set i.e. A := A - {2}. The new A has two minimal elements {1}, and {3}.

Select {1} and remove it from A. Denote by A the resultant set, that is A = {{3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.

This new A has two minimal elements {3} and {1, 2}.

Select {1, 2} and remove it from A.

Proceeding in like manner, we can obtain the following linear order: {{2}, {1}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Discrete structures. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10768/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Discrete structures' conversation and receive update notifications?

Ask