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Partially Ordered Sets. Hasse Diagrams.

“Mr. Smith, could you explain to us what recursion is all about?” The professor in Introduction to Programming asked an apathetic student. “I don’t know the question, but sex, money, or both is definitely the answer, and God, justice, our values, and love are just the excuses,” I replied. “You shall not pass,” the teacher was far from amused, Apocalypse, Anawim, #justtothepoint.

Definition. A partial order $\preceq$ on a set X (written in LaTeX as \preceq) is a binary relation $\preceq$ ⊆ X × X or a subset of the Cartesian product such that is reflexive, antisymmetric, and transitive, ∀x, y, z∈ X

  1. Reflexive: x $\preceq$ x.
  2. Antisymmetric: if x $\preceq$ y and y $\preceq$ x ⇒ x = y.
  3. Transitive: If x $\preceq$ y and y $\preceq$ z, then x $\preceq$ z.

A partially ordered set or poset for short is a set endowed or equipped with a partial order. A set is partially ordered if there is some relation $\preceq$ on the set such that either x $\preceq$ y, y $\preceq$ x, that is, x and y are comparable or x and y are unrelated, e.g., “is father of” and the lexicographical order, that is, (a, b) $\preceq$ (c, d) if a < c or (a = c and b $\preceq$ d).



  1. Reflexive. If x ∈ A ⇒ x ∈ A ⇒ A ⊆ A 😄
  2. Antisymmetric. A ⊆ B and B ⊆ A ⇒ B = A.
  3. Transitive. A ⊆ B and B ⊆ C ⇒ ∀x ∈ A ⇒[A ⊆ B] x ∈ B ⇒[B ⊆ C] ⇒ x ∈ C, therefore A ⊆ C.

The word partial is used to indicate that not every pair of elements needs to be comparable, e.g., let X = {a, b, c, d}, A = {a, b}, B = {c, d}, A ⊈ B and B ⊈ A. A and B are not comparable.

Hasse diagram

A Hasse or lattice diagram is a type of mathematical diagram used to represent a finite partially ordered set (X, $\preceq$). It is a graph whose vertices are the elements of X and for which an edge or segment between two vertices (x, y) exists if they are comparable, x $\preceq$ y and there's no intermediate element that sit between them, that is, whenever x $\preceq$ z $\preceq$ y either z = x or z = y.


We are showing the Hasse diagrams for X = {a, b}, (P(X), ⊆) -Figure 1.a.-, and X = {a, b, c}, (P(X), ⊆) -Figure 1.b.-. Notice ∅ ⊆ {a} ⊆ {a, b} ⊆ X Image 

Besides, the group ℤ4 = ⟨1⟩ = {0, 1, 2, 3} is a cyclic group of order 4 with a completely different Hasse diagram. It has only one proper subgroup, namely {0, 2} = ⟨2⟩. Image 

The dihedral group D4 is the symmetry group of the square. D4 = {id (e), rotations (ρ1 -90° around the center anticlockwise-, ρ2 -180°-, and ρ3 -270°-), reflections or mirror images in the x -horizontal- and y -diagonal- axes (μ1, μ2), and reflections in the diagonals δ1 and δ2 respectively} = {e, r, r2, r3, s, rs, r2s, r3s} = ⟨r, s | r4 = s2 = e, rs = sr3⟩.

Hasse diagram, D₄ 


This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. This post relies heavily on the following resources, specially on NPTEL-NOC IITM, Introduction to Galois Theory, Michael Penn, and Contemporary Abstract Algebra, Joseph, A. Gallian.
  1. NPTEL-NOC IITM, Introduction to Galois Theory.
  2. Algebra, Second Edition, by Michael Artin.
  3. LibreTexts, Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
  4. Field and Galois Theory, by Patrick Morandi. Springer.
  5. Michael Penn (Abstract Algebra), and MathMajor.
  6. Contemporary Abstract Algebra, Joseph, A. Gallian.
  7. Andrew Misseldine: College Algebra and Abstract Algebra.
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