Glossary

2.4 Examples of common lattices

3 min read•august 7, 2024

Lattices come in many shapes and sizes, each with unique properties. Boolean and powerset lattices deal with sets, while diamond and pentagon lattices have specific structures. These examples show how versatile lattices can be in representing different mathematical concepts.

Chain and divisibility lattices introduce order to the mix. Partition lattices organize sets into groups, and geometric lattices connect to vector spaces. These examples highlight how lattices can model various mathematical relationships and structures.

Boolean and Powerset Lattices

Lattices Based on Sets

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    • Consists of all subsets of a finite set ordered by inclusion
    • Has a unique smallest element (empty set) and a unique largest element (the entire set)
    • Number of elements in a Boolean lattice is 2n2^n, where nn is the number of elements in the set
    • Example: The Boolean lattice of the set {a, b, c} has 8 elements: ∅\emptyset, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, and {a, b, c}
    • Formed by the powerset of a set ordered by inclusion
    • Powerset of a set is the set of all subsets of that set
    • Example: The powerset lattice of the set {1, 2} is {∅\emptyset, {1}, {2}, {1, 2}}

Special Lattices with Unique Structures

    • A lattice with four elements: a bottom element, two , and a top element
    • resembles a diamond shape
    • Example: ({0, a, b, 1}, ≤\leq) where 0 ≤\leq a, 0 ≤\leq b, a ≤\leq 1, and b ≤\leq 1
    • A lattice with five elements arranged in a pentagonal shape in the Hasse diagram
    • Has a unique smallest element, a unique largest element, and three incomparable elements in between
    • Example: ({0, a, b, c, 1}, ≤\leq) where 0 ≤\leq a, 0 ≤\leq b, 0 ≤\leq c, a ≤\leq 1, b ≤\leq 1, and c ≤\leq 1

Chain and Divisibility Lattices

Lattices with Total Order

    • A lattice in which any two elements are comparable
    • Also known as a or a linearly ordered set
    • Every pair of elements has a least upper bound () and a greatest lower bound ()
    • Example: The set of natural numbers with the usual order (N\mathbb{N}, ≤\leq) forms a chain lattice
    • A lattice formed by the positive divisors of a positive integer ordered by divisibility
    • The meet of two elements is their (GCD), and the join is their (LCM)
    • Example: The divisibility lattice of 12 is ({1, 2, 3, 4, 6, 12}, ∣\mid) where a∣ba \mid b means "aa divides bb"

Partition and Geometric Lattices

Lattices with Specific Mathematical Properties

    • A lattice formed by the set of all partitions of a set ordered by
    • The meet of two partitions is their coarsest common refinement, and the join is their finest common coarsening
    • Example: The partition lattice of the set {1, 2, 3} has 5 elements: {{1}, {2}, {3}}, {{1, 2}, {3}}, {{1, 3}, {2}}, {{1}, {2, 3}}, and {{1, 2, 3}}
    • A lattice that satisfies the following properties:
      • It is semimodular: if aa and bb cover their meet, then their join covers both aa and bb
      • It is atomistic: every element is a join of atoms (minimal non-zero elements)
    • Often arises from geometric structures like subspaces of a vector space or flats of a matroid
    • Example: The lattice of subspaces of a finite-dimensional vector space forms a geometric lattice

Key Terms to Review (18)

Atomistic Lattice: An atomistic lattice is a specific type of lattice in which every element is an atom or a minimal element. This means that for any element in the lattice, there exists an atom that is less than or equal to it, ensuring that the structure has atoms that serve as the building blocks for all other elements. These lattices are important as they help illustrate concepts such as join and meet operations and play a key role in understanding how elements interact within a lattice structure.
Boolean Lattice: A Boolean lattice is a specific type of lattice that reflects the structure of logical operations, particularly in the context of set theory and propositional logic. It consists of all subsets of a given set, with the join operation corresponding to union and the meet operation corresponding to intersection. This lattice is both complemented and distributive, which means that every element has a unique complement and the operations distribute over each other.
Chain Lattice: A chain lattice is a special type of lattice in which every pair of elements is comparable, meaning that for any two elements, one is either less than or equal to the other. This property results in a linear order among the elements, making it a straightforward and easy-to-understand structure in lattice theory. Chain lattices can serve as examples of more complex lattices and demonstrate fundamental properties such as join and meet operations.
Diamond Lattice: A diamond lattice is a specific type of crystal structure characterized by a three-dimensional arrangement of atoms, where each atom is tetrahedrally coordinated to four other atoms. This arrangement is commonly observed in diamond and silicon, leading to unique properties such as high hardness and excellent electrical insulation. The diamond lattice is an example of a regular lattice that can also be analyzed in the context of modular lattices due to its structural properties and relationships between elements.
Divisibility Lattice: A divisibility lattice is a mathematical structure that represents the relationship between natural numbers based on their divisibility. In this lattice, each element corresponds to a natural number, and the ordering is determined by the divisibility relation: for two elements a and b, a is less than or equal to b if a divides b. This structure showcases the connections between numbers and helps to visualize how they interact through their factors.
Geometric Lattice: A geometric lattice is a specific type of lattice that arises from the relationships among the subsets of a finite set, where the elements can be represented geometrically as points in space. These lattices are often visualized using geometric structures, such as points in Euclidean space, and exhibit properties like being closed under intersection and union. They are important in various mathematical contexts, including combinatorics and topology.
Greatest common divisor: The greatest common divisor (GCD) of two or more integers is the largest positive integer that divides each of the numbers without leaving a remainder. Understanding GCD is essential in various mathematical concepts, particularly in number theory and lattice theory, as it helps in establishing the relationships and structures within sets of numbers. The GCD is significant when examining common divisors in different lattices, such as the integer lattice where elements can be seen as nodes connected by divisibility.
Hasse Diagram: A Hasse diagram is a graphical representation of a finite partially ordered set, which visually depicts the ordering of elements based on their relationships. It simplifies the representation of order relations by omitting transitive edges and displaying only the direct connections between elements, making it easier to visualize concepts like joins and meets.
Incomparable Elements: Incomparable elements in a partially ordered set are pairs of elements that do not have a defined order relationship between them. This means that neither element can be said to be less than or greater than the other, highlighting the structure and complexity of the ordering within the set. Incomparable elements serve to illustrate the limitations of certain ordering relations and are crucial in understanding the nature of partially ordered sets and lattices.
Join: In lattice theory, a join is the least upper bound of a pair of elements in a partially ordered set, meaning it is the smallest element that is greater than or equal to both elements. This concept is vital in understanding the structure of lattices, where every pair of elements has both a join and a meet, which allows for the analysis of their relationships and combinations.
Least Common Multiple: The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of the numbers. This concept is significant in various mathematical contexts, particularly in number theory and lattice theory, where it helps in finding commonalities among different sets and simplifying expressions.
Meet: In lattice theory, the term 'meet' refers to the greatest lower bound (GLB) of a set of elements within a partially ordered set. It identifies the largest element that is less than or equal to each element in the subset, essentially serving as the intersection of those elements in the context of a lattice structure.
Partition Lattice: A partition lattice is a mathematical structure that represents the ways to divide a set into non-empty, disjoint subsets, known as partitions. Each partition can be thought of as a grouping of elements, and the lattice is formed by considering these partitions as nodes connected by relationships defined by set inclusion. The partition lattice showcases the inherent structure of these partitions and how they can be combined or refined, leading to an understanding of their ordering.
Pentagon Lattice: A pentagon lattice is a specific type of lattice structure that has five elements arranged in a way that satisfies the properties of a lattice, including the meet and join operations. This structure can represent relationships and hierarchies in various mathematical contexts and is particularly significant in discussing more complex relationships between elements, such as modularity and distributivity. Its unique arrangement allows for exploration of the nuances between these concepts and serves as a clear example within the broader category of common lattices.
Powerset Lattice: A powerset lattice is the lattice formed by the collection of all subsets of a given set, ordered by inclusion. This structure not only showcases the relationships between the subsets but also highlights key lattice properties, such as bounds and completeness. The powerset lattice illustrates important concepts like meet and join operations, where the meet corresponds to the intersection of subsets and the join corresponds to their union.
Refinement: Refinement refers to the process of making a lattice more precise or structured by introducing finer distinctions among its elements. This concept often highlights how a particular lattice can be broken down into smaller or more detailed sub-lattices, capturing more specific relationships among elements. It can also represent the idea of improving the organization of a lattice, making it easier to analyze and utilize in various applications.
Semimodular Lattice: A semimodular lattice is a type of lattice that satisfies a certain condition regarding its elements, specifically that if one element covers another, then any element that is below the first must also cover an element that is below the second. This concept is crucial for understanding more complex structures in lattice theory and often serves as a bridge between modular lattices and more general lattices.
Totally Ordered Set: A totally ordered set is a type of partially ordered set where every pair of elements is comparable, meaning for any two elements a and b, either a ≤ b or b ≤ a holds true. This concept connects to various structures, helping us understand how elements relate to one another in a complete way. It plays a crucial role in defining common lattices, understanding top and bottom elements, exploring the properties of complete lattices, and analyzing their applications in universal algebra.
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