Elements of POSET | Maximal Element, Minimal Element , Lower and Upper Bounds

Partially Ordered Set (POSET) is a fundamental concept in mathematics and computer science, providing a structured way to analyze and compare elements within a set. In a POSET, not every pair of elements needs to be comparable, making it a versatile tool for representing hierarchical relationships and dependencies. Understanding the elements of a POSET requires knowledge of key concepts such as order relations, minimal and maximal elements, and chains, which are crucial for various applications in fields ranging from data analysis to algorithm design.

In this article we will discuss about POSET, its elements and some key concepts related to it.

POSET (Partially Ordered Set)

A partially ordered set (POSET) is a fundamental concept in mathematics, particularly in order theory and discrete mathematics. A POSET consists of a set combined with a partial order—a binary relation that is reflexive, antisymmetric, and transitive.

A POSET consists of a set P and a binary relation ≤ defined on P which satisfies the Reflexive, Anti-symmetric, and transitive properties.

The elements of a POSET include several key types, such as maximal and minimal elements. A maximal element is not less than any other element in the set, whereas a minimal element is not greater than any other element. Additionally, POSETs can have the greatest and least elements, which are unique elements that are greater than or less than all others, respectively.

In this article, we will discuss about all of the mentioned concepts in detail.

Historical Background and Importance of POSET

This concept or POSETs was formalized in the early 20th century, with contributions from mathematicians exploring order theory and lattice structures. Key development include formation of partial orders and the creation of Hasse diagrams by Helmut Hasse, which usually represent these order relations. The Word partial is used to indicate that not every pair of elements needs to be comparable; that is , there may be pairs for which neither elements precedes the other. So, partial orders thus generalize total orders, in which every pair is comparable. It is a homogeneous binary relation.

Importance of POSET

It is important in Mathematics , Computer Science, Decision Making, Optimization.

In Mathematics POSET provide a framework for studying order and hierarchy in set theory and algebra. In Computer Science It is used in scheduling, task management, and data organization. whereas it also helps in analyzing where choices required with partial comparability. Also , where we don't have complete information and need to make a decision then this will helps to solved those problems.

What is POSET?

POSET, known as Partially Ordered Set , works on the principle of Partial Ordering Relation. A relation R is said to be Partial Ordered Relation when it can satisfy the following properties:

1. R is Reflexive , i.e., if set A ={1,2,3} then R ={(1,1), (2,2), (3,3)} is a Reflexive relation.
2. R is Anti-symmetric , i.e., if R contains (1,2) then (2,1) is not allowed.
3. R is Transitive , i.e., if R contains (1,2), (2,3), then it should contain (1,3) to make it Transitive.

POSET: If A set 'A' following a Partial Ordering Relation 'R' then it is known as POSET. It is denoted by [A; R].

Note- Unlike Asymmetry, Anti-Symmetry allows Reflexive elements like (a, a) or (b, b) in a Relation.

Let's consider an example for better understanding.

Example 1: For a set A = {1,2,3}, check if the following relations are POSET?

R₁ = {(1,1), (2,2), (3,3)}

R₂ = {(1,1), (2,2), (3,3), (1,2), (2,1)}

R₃ = { }

Solution:

Explanation: To prove a Partial Order Relation , check Reflexivity, Anti-Symmetry and Transitivity.

(1,1)	(1,2)	(1,3)
(2,1)	(2,2)	(2,3)
(3,1)	(3,2)	(3,3)

For Relation R ₁ ,

• Reflexive: Since (1,1) (2,2) (3,3) are present so it is Reflexive.
• Anti-symmetry: It allows reflexive pairs, so it is Anti-symmetric.
• Transitive: Reflexive pairs are always Transitive.

For Relation R ₂ ,

• Reflexive: Since (1,1) (2,2) (3,3) are present so it is Reflexive.
• Anti-symmetry: For (1,2) there is (2,1) so not Anti-symmetric.
• Transitive: There are no such pairs (a,b) (b,c) such that (a,c) is not present.

For Relation R ₃ ,

• Reflexive: NULL sets doesn't contain either of (1,1) (2,2) (3,3).

Therefore, R ₁ is a POSET, but R ₂ and R ₃ are not.

Elements of POSET

Some of the common elements of POSET are:

• Maximal Element
• Minimal Element
• Maximum Element (Greatest)
• Minimum Element (Least)
• Upper Bound
• Lower Bound

Maximal Element

If in a POSET/Lattice, an element is not related to any other element. Or, in simple words, it is an element with no outgoing (upward) edge. In the above diagram, A, B, F are Maximal elements.

Minimal Element

If in a POSET/Lattice, no element is related to an element. Or, in simple words, it is an element with no incoming (downward) edge. In the above diagram, C, D, E are Minimal elements.

Maximum Element (Greatest)

If in a POSET/Lattice, it is a Maximal element, and every element is related to it, i.e., every element of the lattice should be connected to this element. In the above diagram, E and F are Maximal elements, but E is the only Maximum element.

Minimum Element (Least)

If in a POSET/Lattice, it is a Minimal element and is related to every other element, i.e., it should be connected to every element of the lattice. In the above diagram, A and B are Minimal elements, but A is the only Minimum element.

Note

• Every Maximum element is a Maximal element but every Maximal element is not a Maximum element
• Every Minimum element is a Minimal element but every Minimal element is not a Minimum element.

Upper Bound

Let's assume B is a subset of set A. An element x ∈ A is in the upper bound of B if (y, x) ∈ POSET where ∀ y ∈ B. Or we can say that it is an element to which every element of a subset is related.

1. B = {E,C}: Upper Bound- {G, E}  ( E can itself be an upper bound because Partial order follows Reflexive property)
2. B = {C,F,D}: Upper Bound- {G, H, F}

Lower Bound

If B is a subset of set A, an element x ∈ A is in the lower bound of B if (x, y) ∈ POSET where ∀ y ∈ B. Or we can say that it is an element that is related /connected to every element of subset B.

1. B = {E, C}: Lower Bound- {A, B, C}  ( C can itself be a lower bound because Partial order follows Reflexive property)
2. B = {C, F, D}: Lower Bound- { ∅ }

Least Upper Bound

Least Upper Bound of a subset S within a POSET is the smallest element in the set that is greater than or equal to every element of S. It is also known as Join. The Minimum (Least) element in Upper Bound.

1. B = {C, D} : Least Upper Bound- { E }
2. B = {A, B} : Least Upper Bound- { D }
3. B = {E, F}  : Least Upper Bound- { ∅ }

Greatest Lower Bound

Greatest Lower Bound of a subset SSS in a POSET is the largest element in the set that is less than or equal to every element of S. It is also known as Meet. The Maximum (Greatest) element in Lower Bound.

1. B = {C,D} : Least Upper Bound- { A }
2. B = {A,B} : Least Upper Bound- { ∅ }
3. B = {E,F}  : Least Upper Bound- { D }

Element of POSET : Solved Examples

Example 1: Identifying elements in a POSET

Consider the POSET (P, ≤) where P = {1, 2, 3, 4} and ≤ is the "divides" relation.

Solution:

Minimal elements: 1 (as it divides no other number except itself)

Maximal elements: 2, 3, 4 (as no number in the set divides them except themselves)

Least element: 1 (as it divides all other numbers)

Greatest element: None (as no number divides all others)

Example 2: Hasse Diagram

Draw the Hasse diagram for the POSET ({1, 2, 3, 4, 6, 12}, |) where | denotes "divides".

Solution:

The Hasse diagram would look like this:

12

/

6 4

| |

3 2

\ /

1

Example 3: Comparable and Incomparable Elements

In the POSET (P(A), ⊆) where A = {a, b, c} and P(A) is the power set of A, identify comparable and incomparable pairs.

Solution:

Comparable pairs: (∅, {a}), (∅, {b}), (∅, {c}), ({a}, {a,b}), ({b}, {a,b}), etc.

Incomparable pairs: ({a}, {b}), ({a}, {c}), ({b}, {c}), ({a,b}, {a,c}), etc.

Example 4: Upper and Lower Bounds

In the POSET ({1, 2, 3, 4, 5, 6}, ≤), find the upper and lower bounds of the subset {2, 3, 5}.

Solution:

Upper bounds: 5, 6

Lower bounds: 1, 2

Least upper bound (supremum): 5

Greatest lower bound (infimum): 2

Example 5: Chains and Antichains

In the POSET (P({a, b, c}), ⊆), identify a maximal chain and a maximal antichain.

Solution:

A maximal chain: ∅ ⊂ {a} ⊂ {a,b} ⊂ {a,b,c}

A maximal antichain: {{a}, {b}, {c}}

Example 6: Lattice Verification

Determine if the POSET ({1,2,3,6,12,18}, |) where | denotes "divides" is a lattice.

Solution:

To be a lattice, every pair of elements must have a least upper bound (LUB) and greatest lower bound (GLB).

Let's check a few pairs:

For 2 and 3: LUB = 6, GLB = 1

For 6 and 12: LUB = 12, GLB = 6

For 2 and 18: LUB = 18, GLB = 2p

Example 7: Dual POSET

Given the POSET P = ({a,b,c,d}, R) where R = {(a,b), (a,c), (b,d), (c,d)}, find its dual POSET.

Solution:

The dual POSET P* = ({a,b,c,d}, R*) where R* is the reverse of all relations in R.

R* = {(b,a), (c,a), (d,b), (d,c)}

The Hasse diagram of P: The Hasse diagram of P*:

a d

/ \ /

b c b c

\ / \ /

d a

Example 8: Linear Extension

Find a linear extension of the POSET ({1,2,3,4}, R) where R = {(1,2), (1,3), (2,4), (3,4)}.

Solution:

A linear extension is a total order that respects the partial order.

One possible linear extension is: 1 < 2 < 3 < 4

Another valid linear extension is: 1 < 3 < 2 < 4

Example 9: Complement in a Boolean Algebra

In the Boolean algebra of subsets of {1,2,3}, find the complement of {1,3}.

Solution:

The universe U = {1,2,3}

The complement of {1,3} is U \ {1,3} = {2}

Example 10: Join and Meet in a Lattice

In the lattice of divisors of 30, find:

a) 6 ∨ 10 (join)

b) 6 ∧ 10 (meet)

Solution:

Divisors of 30: {1, 2, 3, 5, 6, 10, 15, 30}

a) 6 ∨ 10 = LUB(6,10) = 30

b) 6 ∧ 10 = GLB(6,10) = 2

Also read,

• Partial Orders and Lattices
• Types of Sets in Discrete Mathematics
• Group Theory

Conclusion

In conclusion, understanding the elements of a partially ordered set (POSET) is crucial for grasping the broader concepts in order theory and discrete mathematics. A POSET is defined by its partial order relation, which is reflexive, antisymmetric, and transitive. The key elements, such as maximal and minimal elements, upper and lower bounds, and greatest and least elements, help in analyzing the structure and properties of sets with partial orders.