7.1 Complemented lattices and their properties
3 min read•august 7, 2024
Complemented lattices are bounded lattices where every element has a . They're key to understanding Boolean algebras and set theory. These lattices have special properties like the and De Morgan's laws.
There are different types of complemented lattices, each with unique features. Orthocomplemented lattices have unique complements, while distributive complemented lattices combine complementation with distributivity. Pseudocomplemented lattices introduce a weaker notion of complementation.
Complements and Complemented Lattices
Definition and Uniqueness of Complements
Top images from around the web for Definition and Uniqueness of Complements
How to draw simple lattice diagrams (MathJax syntax) - Mathematics Meta Stack Exchange View original
Is this image relevant?
Complemented lattice - Wikipedia View original
Is this image relevant?
How to draw simple lattice diagrams (MathJax syntax) - Mathematics Meta Stack Exchange View original
Is this image relevant?
Complemented lattice - Wikipedia View original
Is this image relevant?
1 of 2
Top images from around the web for Definition and Uniqueness of Complements
How to draw simple lattice diagrams (MathJax syntax) - Mathematics Meta Stack Exchange View original
Is this image relevant?
Complemented lattice - Wikipedia View original
Is this image relevant?
How to draw simple lattice diagrams (MathJax syntax) - Mathematics Meta Stack Exchange View original
Is this image relevant?
Complemented lattice - Wikipedia View original
Is this image relevant?
1 of 2
- is a in which every element has a complement
- Bounded lattice contains a top element and a bottom element
- Example: The lattice of subsets of a set is complemented ()
- Complement of an element in a bounded lattice is an element such that:
- ( of and equals the top element)
- ( of and equals the bottom element)
- Uniqueness of complements
- In a complemented lattice, complements are not necessarily unique
- Example: In the lattice (pentagon), the element has two complements, and
Properties of Complemented Lattices
- Double complement law: In a complemented lattice, the double complement of an element is the element itself
- If is a complement of , then is a complement of
- Denoted as , where is a complement of
- : The complement operation in a complemented lattice is an involution
- An involution is a function such that for all in the domain
- De Morgan's laws hold in complemented lattices
- (complement of join equals meet of complements)
- (complement of meet equals join of complements)
Special Types of Complemented Lattices
Orthocomplemented Lattices
- is a complemented lattice satisfying an additional axiom:
- implies ( of complement)
- In an orthocomplemented lattice, complements are unique
- If has complements and , then
- Example: The lattice of closed subspaces of a Hilbert space is orthocomplemented
Distributive Complemented Lattices
- is a complemented lattice that is also distributive
- Distributivity: and
- In a distributive complemented lattice, complements are unique
- Example: The lattice of subsets of a set (power set lattice) is a distributive complemented lattice
Pseudocomplemented Lattices
- is a bounded lattice in which every element has a
- Pseudocomplement of an element is the greatest element such that
- Denoted as , where is the pseudocomplement of
- In a pseudocomplemented lattice, the pseudocomplement may not be a true complement
- may not equal the top element
- Example: The lattice of ideals of a ring is a pseudocomplemented lattice
Key Terms to Review (13)
Bounded lattice: A bounded lattice is a specific type of lattice that contains both a least element (often denoted as 0) and a greatest element (often denoted as 1). This structure allows for every pair of elements to have a unique least upper bound (join) and greatest lower bound (meet), making it fundamental in various mathematical contexts.
Complement: In lattice theory, a complement of an element in a lattice is another element that, when combined with the original element using the join operation, yields the greatest element (often denoted as 1), and when combined using the meet operation, yields the least element (often denoted as 0). This concept is crucial for understanding structures like Boolean algebras and distributive lattices, where every element has a unique complement.
Complemented Lattice: A complemented lattice is a type of lattice where every element has a complement, meaning for any element 'a', there exists an element 'b' such that the meet (greatest lower bound) of 'a' and 'b' is the minimum element, and the join (least upper bound) of 'a' and 'b' is the maximum element. This property is crucial in understanding the structure and behavior of various lattices, as it directly connects to concepts like modularity and distributivity, influencing both theoretical applications and practical uses in algebraic structures.
Distributive complemented lattice: A distributive complemented lattice is a type of lattice in which every pair of elements satisfies the distributive property and every element has a complement. In this structure, for any two elements a and b, the operation satisfies a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) and there exists an element x such that a ∨ x = 1 and a ∧ x = 0. This definition is crucial in understanding the properties and applications of complemented lattices.
Double complement law: The double complement law states that for any element in a complemented lattice, taking the complement of the complement of that element returns the original element. This can be expressed mathematically as $$x'' = x$$, where $$x$$ is an element of the lattice. This law highlights a fundamental property of complemented lattices, emphasizing their structure and the interplay between elements and their complements.
Involution: In lattice theory, an involution is a special type of unary operation on a lattice that satisfies the property of being its own inverse. This means that applying the operation twice will return the original element, which is expressed mathematically as \( f(f(x)) = x \) for all elements \( x \) in the lattice. In the context of complemented lattices, involutions are crucial as they help define the concept of complements and facilitate the understanding of lattice properties such as duality and symmetry.
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.
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.
Order-Reversing Property: The order-reversing property refers to a characteristic of certain mappings between partially ordered sets where the order of elements is reversed. If a function satisfies this property, it means that if one element is less than another in the original set, its image under the function will be greater than or equal to the image of the larger element. This property is particularly relevant in complemented lattices, where it helps in understanding how complements and order structures interact within the lattice.
Orthocomplemented lattice: An orthocomplemented lattice is a specific type of lattice where every element has a unique orthocomplement. This means that for every element 'a' in the lattice, there exists an element 'a⊥' such that certain conditions hold: 'a ∨ a⊥' is the greatest element and 'a ∧ a⊥' is the least element. This structure provides a way to define concepts of orthogonality and complements within the framework of lattice theory, linking it to important properties such as modularity and distributivity.
Power Set Lattice: A power set lattice is a specific type of lattice formed by the collection of all subsets of a given set, ordered by inclusion. This structure is crucial as it illustrates fundamental properties of lattices, such as the existence of top and bottom elements, complete lattices, complemented lattices, and how these concepts apply in various domains like logic and programming semantics.
Pseudocomplement: A pseudocomplement in a lattice is an element that acts as a sort of 'complement' but only under certain conditions. Specifically, for an element $a$ in a lattice $L$, the pseudocomplement, often denoted as $a^*$, is the greatest element $b$ such that $a igvee b$ is less than or equal to the top element of the lattice, usually denoted by $1$. This concept helps in understanding the structure of complemented lattices and illustrates properties that are vital when exploring more complex lattice behaviors.
Pseudocomplemented lattice: A pseudocomplemented lattice is a type of lattice in which every element has a pseudocomplement. This means that for any element 'a' in the lattice, there exists an element 'b' such that 'a' meets 'b' is the least upper bound of all elements that are less than or equal to 'a'. This property ties into the characteristics of complemented lattices, where every element has both a complement and a pseudocomplement.