In ring theory the Hasse diagram of ideals ordered by inclusion is used often. In particular the attached Moebius function is used to compute the so-called homogenous weight in Coding Theory.

In ring theory the Hasse diagram of ideals ordered by inclusion is used often. In particular the attached Moebius function is used to compute the so-called homogenous weight in Coding Theory.

I don't know what exactly counts as an "application".

The Jordan-Hölder theorem of group theory is essentially a theorem about paths in the Hasse diagram of subgroups (though the ordering is not exactly inclusion, but rather the transitive closure of the "normal subgroup" relation).

This is generalized by the Jordan-Hölder-Dedekind theorem in semi-modular lattices, that says that paths in the Hasse diagram of a semi-modular lattice with the same endpoints have the same length, and are also are related by a relation of "projectivity".

From this theorem follows the Fundamental Theorem of Projective Geometry: that two finite-dimensional vector space over fields (or skew fields) with the same lattice of subspaces have the same dimension and the same base field (up to isomorphism, of course) provided the dimension of at least one of same is greater than 2.

See chapter 8 of Jacobson / Basic Algebra I.

https://books.google.co.il/books?id=JHFpv0tKiBAC&lpg=PP1&dq=Jacobson%20%2F%20Basic%20Algebra%20I.&pg=PA466#v=onepage&q&f=false

Hi!

In formal language theory (and most probably in other fields of discrete mathematics), hierarchies of languages (or other types of sets) are of high importance. In several papers, as a summary of previous knowledge, or summary of the new results, Hasse diagrams are used to present the relations of the language classes under (set theoretic inclusion). These pictures also help to "memorize" the results, since the information is provided in a graphical way (and diagrams sometimes offer a much faster way of learning/understanding than words).

I use the Hasse Diagram of the symmetries of a square to organize the pattern of the types of quadrilaterals in the plane.  This is an insight I learned from John Conway - a way of organizing a jumble of different definitions into a larger pattern that supports inclusive reasoning, and identifying which symmetries are central to a lot of reasoning. See the attached files.  This also supports the shift towards reasoning with modern geometric approaches (transformations).  BTW, this approach also works on the sphere!

This is a very good question with lots of possible answers.

There are quite papers, figures on Hasse diagrams and their applications, available on RG.  Here are some examples:

https://www.researchgate.net/figure/264167263_fig3_The-Hasse-diagram-of

https://www.researchgate.net/publication/228453365_Hasse_Diagram_Technique-A_Useful_Tool_for_Life_Cycle_Assessment_of_Refrigerants

https://www.researchgate.net/publication/43350606_Ranking_Using_the_Copeland_Score_A_Comparison_with_the_Hasse_Diagram

https://www.researchgate.net/publication/11006805_Application_of_a_Sediment_Quality_Triad_and_Different_Statistical_Approaches_Hasse_Diagrams_and_Fuzzy_Logic_for_the_Comparative_Evaluation_of_Small_Streams

https://www.researchgate.net/publication/236848462_A_Novel_Similarity_Measure_for_Sequence_Data

Article Hasse Diagram Technique–A Useful Tool for Life Cycle Assessm...

Article Ranking Using the Copeland Score: A Comparison with the Hasse Diagram

Article Application of a Sediment Quality Triad and Different Statis...

Article A Novel Similarity Measure for Sequence Data

Dear Dr.

The following papers are useful

http://www.springer.com/cda/content/document/cda_downloaddocument/9781441984760-c1.pdf?SGWID=0-0-45-1157742-p174095771

www.cs.rutgers.edu/~elgammal/classes/.../chapt76.pdf

www.dsi.unive.it/~avp/03_AVP_2015.pdfC