Before starting, I suppose that you don’t know what is lattice so i will explain lattice to you then i will give you explanation of how to identify if the hasse diagram or graph is lattice or not.
Definition: The poset(S, R) is called a lattice iff it is a meet semilattice and a join semilattice.
Now what the definition says, So the definition says if a graph consist of both meet semilattice and join semilattice than the graph will be called as lattice.
What is Meet Semilattice and Join Semilattice
In this above diagram you can see edges of graph, if they only meet in the end at single point then it will be called as meet semilattice. Like in this graph d and d meets at 0 in bottom. so it is a meet semilattice.
If we talk about upper half then it will be called join semilattice, because d and f are meeting each other at a in top.
So in summary, if two edges in a graph meets only in end and not on top then it is called meet semilattice. If two edges in a graph meets only at top and not at the end then it is called join semilattice.
In this figure, you can see that it consists of both meet semilattice and join semilattice, So it will be called a lattice.
This is about lattice. yeah, that’s very simple topic in discrete mathematics.
What is PoSet
A Partially Ordered Set (POSET) is a fundamental concept in mathematics and computer science, 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.
Key Properties of POSET
Transitivity: If (a \leq b) and (b \leq c), then (a \leq c).
Reflexivity: Every element is related to itself. For example, if (a) is an element of the set, then (a \leq a).
Antisymmetry: If (a \leq b) and (b \leq a), then (a = b).
Elements of POSET
- Maximal Element: An element that is not less than any other element in the set. For example, in a set with elements (A, B, F), these could be maximal elements.
- Minimal Element: An element that no other element is less than. For example, in a set with elements (C, D, E), these could be minimal elements.
- Greatest Element: An element that is greater than or equal to every other element in the set.
- Least Element: An element that is less than or equal to every other element in the set.
- Upper Bound: An element (x) is an upper bound of a subset (B) if every element of (B) is less than or equal to (x).
- Lower Bound: An element (x) is a lower bound of a subset (B) if every element of (B) is greater than or equal to (x).
Examples
- Divisibility: The set of natural numbers ordered by divisibility is a POSET. For instance, 1 is a least element as it divides all other numbers, but there is no greatest element.
- Power Set: The set of all subsets of a given set ordered by inclusion is a POSET. For example, the power set of ({a, b, c}) ordered by inclusion forms a POSET.
Hasse Diagrams
A Hasse Diagram is a graphical representation of a POSET. It shows the elements as nodes and the order relations as edges, omitting reflexive and transitive edges for simplicity.
Applications
POSETs are used in various fields such as:
- Mathematics: Studying order and hierarchy in set theory and algebra.
- Computer Science: Scheduling, task management, and data organization.
- Decision Making: Analyzing choices with partial comparability.
Understanding POSETs is crucial for grasping broader concepts in order theory and discrete mathematics. They provide a structured way to analyze and compare elements within a set, making them a versatile tool for representing hierarchical relationships and dependencies
Steps to Identify a Lattice
Step 1: Check if the Set is a Poset
Ensure that the given set PPP with the relation ≤\leq≤ satisfies:
- Reflexivity: A Set which have self pairs of elements is known as reflexive relation of the set.
- Antisymmetry: A Set which do not have symmetrical elements are known as Antisymmetry relation like (2,3) so there should not be (3, 2).
- Transitivity: If in a set there exists {x, y}, {y, z} then there should be present (x, z).
Step 2: Find the Least Upper Bound (LUB) for Every Pair
For each pair of elements a,b∈Pa, b \in Pa,b∈P, check if a supremum (join, a∨ba \vee ba∨b) exists, i.e.,
- a∨ba \vee ba∨b should be the smallest element greater than or equal to both aaa and bbb.
Step 3: Find the Greatest Lower Bound (GLB) for Every Pair
For each pair a,b∈Pa, b \in Pa,b∈P, check if an infimum (meet, a∧ba \wedge ba∧b) exists, i.e.,
- a∧ba \wedge ba∧b should be the largest element smaller than or equal to both aaa and bbb.
Step 4: Verify for All Pairs
If every pair in PPP has both a join (LUB) and a meet (GLB), then the poset is a lattice.
Step 5: (Optional) Identify Special Lattices
- Distributive Lattice: If a∧(b∨c)=(a∧b)∨(a∧c)a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)a∧(b∨c)=(a∧b)∨(a∧c) and a∨(b∧c)=(a∨b)∧(a∨c)a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)a∨(b∧c)=(a∨b)∧(a∨c) for all elements.
- Modular Lattice: If a≤ca \leq ca≤c implies a∨(b∧c)=(a∨b)∧ca \vee (b \wedge c) = (a \vee b) \wedge ca∨(b∧c)=(a∨b)∧c.
- Boolean Algebra: If the lattice has complements (i.e., each element has a unique complement).
Example
Consider P={1,2,3,6}P = \{1, 2, 3, 6\}P={1,2,3,6} with divisibility as the relation:
- LUB of (2,3)=6\text{LUB of } (2,3) = 6LUB of (2,3)=6, GLB = 1
- LUB of (2,6)=6\text{LUB of } (2,6) = 6LUB of (2,6)=6, GLB = 2
- LUB of (3,6)=6\text{LUB of } (3,6) = 6LUB of (3,6)=6, GLB = 3
Since all pairs have both LUB and GLB, PPP is a lattice.