Hasse diagram

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

Misc.

Subsets of a 2-element set			Subsets of a 3-element set			Divisors of 60 ordered by divisibility			Non-negative integers ordered by divisibility
Young–Fibonacci lattice			Young's lattice			Left: Divisors of 120 ordered by divisibility (Birkhoff's representation theorem)
Tamari lattice of order 4 (Associahedron of order 4)				Permutohedron of order 4				Lattice of regular bands
Free distributive lattices of monotonic Boolean functions							Rieger–Nishimura lattice (free Heyting algebra over one generator)

Tesseract

Subsets of a 4-element set:

Emphasize on two cubes	Rhombic dodecahedral parallel projection of the tesseract	Logical connectives
Emphasize on all eight cubes	4x4 matrix	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

Partitions of a 4-element set ordered by refinement:

		Only the 14 noncrossing partitions (This diagram is also vertically symmetric.)
Emphasize on sublattice	Emphasize on symmetry	Emphasize on number of elements per rank

Lattice of subgroups

Symmetric group S4	Dihedral group Dih4	Z23
Z24 This is an untypical Hasse diagram because some elements are displayed below elements that actually have a lower rank, but only when the elements are not linked by an edge. As the rank can not be derived from the position in the Hasse diagram it is expressed in the background color. View large (2048 px wide PNG)        View very large (2,80 m wide SVG)

First-order logic

The eight different meanings that can be expressed with a binary relation and quantifiers, ordered by logical implication:
The only difference between this graph and a cube is the edge in the middle. It makes that these Hasse diagrams don't represent a lattice.

Properties of the logical matrices	Sets of 2x2 logical matrices
Sets of 3x3 logical matrices

46 different meanings can be expressed with a ternary relation and quantifiers.
This is the lattice of the 26 of them that correspond to logical tensors that can be described without referring to diagonals:

Abbreviated descriptions of the logical tensors on the left