Hasse diagram

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A Hasse diagram is a graphical representation of a partially ordered set.


Inclusion ordering.svg
Subsets of a 2-element set
Hasse diagram of powerset of 3; bits.svg
Subsets of a 3-element set
Lattice of the divisibility of 60; factors.svg
Divisors of 60 ordered by divisibility
Infinite lattice of divisors.svg
Non-negative integers ordered by divisibility
Young–Fibonacci lattice
Young's lattice.svg
Young's lattice
Left: Divisors of 120 ordered by divisibility
(Birkhoff's representation theorem)
Tamari lattice, ovals.svg
Associahedron of order 4
Symmetric group 4; permutohedron 3D; l-e factorial numbers.svg
Permutohedron of order 4
Lattice of regular bands
Monotone Boolean functions 0,1,2,3.svg
Free distributive lattices
of monotonic Boolean functions
Rieger–Nishimura lattice
(free Heyting algebra over one generator)
Quadrilateral hierarchy.png
Types of quadriliterals


Subsets of a 4-element set:

Hypercubecubes binary.svg
Emphasis on two cubes
Hypercubeorder binary.svg
Rhombic dodecahedral
parallel projection of the tesseract
Logical connectives Hasse diagram.svg
Logical connectives
Tesseract Hasse diagram with nibble shorthands.svg
Emphasis on all eight cubes
Tesseract Hasse diagram with nibble shorthands; like 4x4 matrix.svg
4x4 matrix
Tesseract tetrahedron shadow matrices.svg
Tetrahedral central projection of the tesseract
Not a Hasse diagram, but similar: Highest element in center;
lower elements farer away from center; lowest element not shown

Set partitions[edit]

Partitions of a 4-element set ordered by refinement:

Set partitions 4; Hasse; matrices.svg
Set partitions 4; Hasse; circles.svg
Noncrossing partitions 4; Hasse.svg

Only the 14 noncrossing partitions
(This diagram is also vertically symmetric.)
Set partitions 4; Hasse sub; numbers.svg
Emphasis on sublattice
Set partitions 4; Hasse; numbers.svg
Emphasis on symmetry
Set partitions 4; Hasse flat; numbers.svg
Emphasis on number of elements per rank

Lattice of subgroups[edit]

Dih4 subgroups (cycle graphs).svg
Dihedral group Dih4
Z2^3; Lattice of subgroups Hasse diagram.svg
Z2^4; Lattice of subgroups Hasse diagram.svg
Z24 (rank expressed by background color, not by position)
Symmetric group S4; lattice of subgroups Hasse diagram; all 30 subgroups.svg
30 subgroups of S4
Full octahedral group; subgroups Hasse diagram; rotational.svg
9 types of subgroups of S4
Full octahedral group; subgroups Hasse diagram.svg
25 types of subgroups of the S4 × C2

First-order logic[edit]

v:Formulas in predicate logic

Cocoon concertina square graph; matrix sketches.svg Concertina square graph; matrix sketches.svg Concertina square graph.svg Concertina cube graph.svg Concertina tesseract graph.svg

Root systems[edit]