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:
- R is Reflexive , i.e., if set A ={1,2,3} then R ={(1,1), (2,2), (3,3)} is a Reflexive relation.
- R is Anti-symmetric , i.e., if R contains (1,2) then (2,1) is not allowed.
- 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
Elements of POSET 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.
- B = {E,C}: Upper Bound- {G, E} ( E can itself be an upper bound because Partial order follows Reflexive property)
- 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.
- B = {E, C}: Lower Bound- {A, B, C} ( C can itself be a lower bound because Partial order follows Reflexive property)
- 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.
- B = {C, D} : Least Upper Bound- { E }
- B = {A, B} : Least Upper Bound- { D }
- 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.
- B = {C,D} : Least Upper Bound- { A }
- B = {A,B} : Least Upper Bound- { ∅ }
- 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,
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.
Similar Reads
Engineering Mathematics Tutorials Engineering mathematics is a vital component of the engineering discipline, offering the analytical tools and techniques necessary for solving complex problems across various fields. Whether you're designing a bridge, optimizing a manufacturing process, or developing algorithms for computer systems,
3 min read
Linear Algebra
MatricesMatrices are key concepts in mathematics, widely used in solving equations and problems in fields like physics and computer science. A matrix is simply a grid of numbers, and a determinant is a value calculated from a square matrix.Example: \begin{bmatrix} 6 & 9 \\ 5 & -4 \\ \end{bmatrix}_{2
3 min read
Row Echelon FormRow Echelon Form (REF) of a matrix simplifies solving systems of linear equations, understanding linear transformations, and working with matrix equations. A matrix is in Row Echelon form if it has the following properties:Zero Rows at the Bottom: If there are any rows that are completely filled wit
4 min read
Eigenvalues and EigenvectorsEigenvalues and eigenvectors are fundamental concepts in linear algebra, used in various applications such as matrix diagonalization, stability analysis and data analysis (e.g., PCA). They are associated with a square matrix and provide insights into its properties.Eigen value and Eigen vectorTable
10 min read
System of Linear EquationsA system of linear equations is a set of two or more linear equations involving the same variables. Each equation represents a straight line or a plane and the solution to the system is the set of values for the variables that satisfy all equations simultaneously.Here is simple example of system of
5 min read
Matrix DiagonalizationMatrix diagonalization is the process of reducing a square matrix into its diagonal form using a similarity transformation. This process is useful because diagonal matrices are easier to work with, especially when raising them to integer powers.Not all matrices are diagonalizable. A matrix is diagon
8 min read
LU DecompositionLU decomposition or factorization of a matrix is the factorization of a given square matrix into two triangular matrices, one upper triangular matrix and one lower triangular matrix, such that the product of these two matrices gives the original matrix. It is a fundamental technique in linear algebr
6 min read
Finding Inverse of a Square Matrix using Cayley Hamilton Theorem in MATLABMatrix is the set of numbers arranged in rows & columns in order to form a Rectangular array. Here, those numbers are called the entries or elements of that matrix. A Rectangular array of (m*n) numbers in the form of 'm' horizontal lines (rows) & 'n' vertical lines (called columns), is calle
4 min read
Sequence & Series
Calculus
Limits, Continuity and DifferentiabilityLimits, Continuity, and Differentiation are fundamental concepts in calculus. They are essential for analyzing and understanding function behavior and are crucial for solving real-world problems in physics, engineering, and economics.Table of ContentLimitsKey Characteristics of LimitsExample of Limi
10 min read
Cauchy's Mean Value TheoremCauchy's Mean Value theorem provides a relation between the change of two functions over a fixed interval with their derivative. It is a special case of Lagrange Mean Value Theorem. Cauchy's Mean Value theorem is also called the Extended Mean Value Theorem or the Second Mean Value Theorem.According
7 min read
Taylor SeriesA Taylor series represents a function as an infinite sum of terms, calculated from the values of its derivatives at a single point.Taylor series is a powerful mathematical tool used to approximate complex functions with an infinite sum of terms derived from the function's derivatives at a single poi
8 min read
Inverse functions and composition of functionsInverse Functions - In mathematics a function, a, is said to be an inverse of another, b, if given the output of b a returns the input value given to b. Additionally, this must hold true for every element in the domain co-domain(range) of b. In other words, assuming x and y are constants, if b(x) =
3 min read
Definite Integral | Definition, Formula & How to CalculateA definite integral is an integral that calculates a fixed value for the area under a curve between two specified limits. The resulting value represents the sum of all infinitesimal quantities within these boundaries. i.e. if we integrate any function within a fixed interval it is called a Definite
8 min read
Application of Derivative - Maxima and MinimaDerivatives have many applications, like finding rate of change, approximation, maxima/minima and tangent. In this section, we focus on their use in finding maxima and minima.Note: If f(x) is a continuous function, then for every continuous function on a closed interval has a maximum and a minimum v
6 min read
Probability & Statistics
Mean, Variance and Standard DeviationMean, Variance and Standard Deviation are fundamental concepts in statistics and engineering mathematics, essential for analyzing and interpreting data. These measures provide insights into data's central tendency, dispersion, and spread, which are crucial for making informed decisions in various en
10 min read
Conditional ProbabilityConditional probability defines the probability of an event occurring based on a given condition or prior knowledge of another event. It is the likelihood of an event occurring, given that another event has already occurred. In probability, this is denoted as A given B, expressed as P(A | B), indica
12 min read
Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba
13 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows
13 min read
Covariance and CorrelationCovariance and correlation are the two key concepts in Statistics that help us analyze the relationship between two variables. Covariance measures how two variables change together, indicating whether they move in the same or opposite directions. Relationship between Independent and dependent variab
6 min read
Practice Questions