“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
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).
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.
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.
Examples:
Because of the transitivity property, any path upward the graph shows comparability of subsets.
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
Figure 1.c. shows the Hasse diagram for A = {2, 3, 4, 7, 8, 12, 14, 24, 35}, and R the divides relation on A = {(x, y) ∈ A x A: x | y}
The Hasse diagram for the subgroups of ℤ_{6} is shown in Figure 1.d
ℤ_{2}⊕ℤ_{2} = {(0, 0), (1, 0), (0, 1), (1, 1)}. |ℤ_{2} ⊕ ℤ_{2}| = 4 = 2p, p=2, 2 prime, and it is not cyclic ⇒ [Classification of Groups of Order 2p] G is either isomorphic to ℤ_{4} (cyclic ⊥) or the dihedral group D_{2}. Therefore, ℤ_{2} ⊕ ℤ_{2} ≋ D_{2} ≋ K_{4}, the Klein-four-group. Its Hasse diagrams is shown in Figure 1.e.
Besides, the group ℤ_{4} = ⟨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⟩.
The dihedral group D4 is the symmetry group of the square. D_{4} = {id (e), rotations (ρ_{1} -90° around the center anticlockwise-, ρ_{2} -180°-, and ρ_{3} -270°-), reflections or mirror images in the x -horizontal- and y -diagonal- axes (μ_{1}, μ_{2}), and reflections in the diagonals δ_{1} and δ_{2} respectively} = {e, r, r^{2}, r^{3}, s, rs, r^{2}s, r^{3}s} = ⟨r, s | r^{4} = s^{2} = e, rs = sr^{3}⟩.