Skip to main content
Logo image

Section 4.4 Partially Ordered Sets

Subsection 4.4.1 Terminology

Definition 4.4.1. Poset.

A relation is an partially ordered set (poset) if and only if the relation is reflexive, anti-symmetric, and transitive.
Consider the following poset defined by listing the relations explicitly.
\(R=\{(a,a), \) \((a,d), \) \((a,f), \) \((a,g), \) \((a,h),\) \((b,b), \) \((b,d), \) \((b,e), \) \((b,f), \) \((b,g), \) \((b,h),\) \((c,c), \) \((c,d), \) \((c,f), \) \((c,g), \) \((c,h),\) \((d,d), \) \((d,f), \) \((d,g), \) \((d,h),\) \((e,e), \) \((e,g), \) \((e,h),\) \((f,f), \) \((g,g), \) \((h,h) \}.\)
Another way to define the elements of the relation is a Hasse diagram. FigureΒ 4.4.2 shows the Hasse diagram for \(R.\) The rules for constructing these diagrams are in ListΒ 4.4.3.
Figure 4.4.2. Hasse Diagram
List 4.4.3. Constructing a Hasse Diagram
  • If \((a,b) \in R\) then \(a\) is below \(b\) on the diagram.
  • If \((a,b) \in R\) then there is a line from \(a\) to \(b\)
    • unless \(a=b\) (reflexive), or
    • unless \((a,c), (c,b) \in R\) (transitive set).
Note that constructing a Hasse diagram requires sorting the elements of the set. Each level of the diagram in some sense is β€œbefore” the levels above it. Because of this ordering in posets we write \(a \preceq b\) to mean \(aRb\text{.}\)

Subsection 4.4.2 Practice

Checkpoint 4.4.4.

Determine whether the following relation is a poset. Let the relation \(R\) on the set of numbers \(\{1,2,3,6\}\) be defined by \(aRb\) if and only if \(a|b.\)

Checkpoint 4.4.5.

Determine whether the following relation is a poset. Let the relation \(R\) on the set of numbers \(\{1,2,3,5,6,10,15,30\}\) be defined by \(aRb\) if and only if \(a|b.\)

Checkpoint 4.4.6.

Determine whether the following relation is a poset. Let the relation \(R\) on the set of positive integers be defined by \(aRb\) if and only if \(a|b.\)

Checkpoint 4.4.7.

Determine whether the following relation is a poset. Let the relation \(R\) on the set of real numbers be defined by \(aRb\) if and only if \(a \le b.\)

Checkpoint 4.4.8.

Determine whether the following relation is a poset. Consider the set \(\Pow(X)\) for the set \(X=\{a,b,c\}.\) Let the relation \(R\) be defined by \(A\) is related to \(B\) if and only if \(A \subseteq B\text{.}\)

Checkpoint 4.4.9.

Draw the Hasse diagrams for the following posets.
  1. \(\subseteq\) on \(\Pow(\{0,1\}).\) Note it may help to list all the elements first.
  2. \(|\) (i.e., \(aRb\) iff \(a|b\)) on \(\{1,2,3,6\}.\)
  3. \(|\) on \(\{1,2,3,5,6,10,15,30\}.\)
  4. \(X=\{ \)(0,0,0), (1,0,0), (0,1,0), (0,0,1), (0,1,1), (1,0,1), (1,1,0), (1,1,1) \(\}\) with the relation \((a_1,a_2,a_3) \preceq (b_1,b_2,b_3)\) iff \(a_i \le b_i\) for all \(i.\)

Subsection 4.4.3 Poset Properties

Subsubsection 4.4.3.1 Terminology

Definition 4.4.10. Comparable.
Two elements \(a,b\) of a poset are comparable if and only if either \(a \preceq b\) or \(b \preceq a.\)
Example 4.4.11. Comparable Elements.
For the poset represented by the Hasse diagram in FigureΒ 4.4.2 the elements d and g are comparable. The elements b and f are incomparable.
Definition 4.4.12. Total Ordering.
A partially ordered set is a totally ordered set if and only if all pairs of elements are comparable.
Definition 4.4.13. Least Upper Bound.
An element \(c\) is a least upper bound of two elements \(a\) and \(b\text{,}\) denoted \(a \wedge b\text{,}\) if and only if \(a \preceq c,\) \(b \preceq c\) and if \(a \preceq d,\) \(b \preceq d\) for any other element then \(c \preceq d.\)
Example 4.4.14. LUB.
For the poset represented by the Hasse diagram in FigureΒ 4.4.2 \(lub(a,b)=d\text{.}\) Note h is another upper bound of a and b, but \(d \preceq h\text{.}\)
Definition 4.4.15. Greatest Lower Bound.
An element \(c\) is a greatest lower bound of two elements \(a\) and \(b\text{,}\) denoted \(a \vee b\text{,}\) if and only if \(c \preceq a,\) \(c \preceq b\) and if \(d \preceq a,\) \(d \preceq b\) for any other element then \(d \preceq c.\)
Example 4.4.16. GLB.
For the poset represented by the Hasse diagram in FigureΒ 4.4.2 \(glb(d,e)=b\text{.}\) There are no other lower bounds for this pair.
Definition 4.4.17. Minimal Element.
An element \(a\) in a poset is a minimal element if and only if there does not exist an element \(b\) such that \(b \preceq a.\)
Example 4.4.18. Minimal Element.
For the poset represented by the Hasse diagram in FigureΒ 4.4.2 the minimal elements are a,b, and c. a is a minimal element because there is no element x such that \(x \preceq a\text{.}\) Notice where the minimal elements are in the Hasse diagram.
Definition 4.4.19. Maximal Element.
An element \(a\) in a poset is a maximal element if and only if there does not exist an element \(b\) such that \(a \preceq b.\)
Example 4.4.20. Maximal Element.
For the poset represented by the Hasse diagram in FigureΒ 4.4.2 the maximal elements are f,g, and h. f is a maximal element because there is no element x such that \(f \preceq x\text{.}\)

Subsubsection 4.4.3.2 Practice

Definition 4.4.25. Lattice.
A poset in which every pair of elements has both a least upper bound and a greatest lower bound is a lattice
Checkpoint 4.4.27.
Why is the name β€œlattice” appropriate?