Bounded and Complemented Lattices

Definition

A bounded lattice is a lattice $(L, \leq)$ that has:

• a least element $0$ such that $0 \leq x$ for all $x \in L$,
• a greatest element $1$ such that $x \leq 1$ for all $x \in L$.

A complemented lattice is a bounded lattice in which every element $a \in L$ has at least one element $b \in L$ called a complement of $a$, satisfying:

• $a \wedge b = 0$,
• $a \vee b = 1$.

Here:

• $\wedge$ means meet,
• $\vee$ means join.

If the complement of every element is unique, the lattice is called a uniquely complemented lattice. If, in addition, the lattice satisfies distributive laws, it becomes a Boolean lattice.

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Main Content

1. Bounded Lattice

• A lattice is bounded when it contains both an absolute bottom element and an absolute top element.
• These elements are denoted by 0 and 1, and they play a crucial structural role because they provide global reference points for the entire lattice.

In a bounded lattice:

• $0$ is the least element, so for every $a \in L$, $0 \le a$.
• $1$ is the greatest element, so for every $a \in L$, $a \le 1$.

This means every element of the lattice lies “between” 0 and 1.

Example 1: Power set lattice

Let $S = \{a,b\}$. The collection of all subsets of $S$ is: $$\mathcal{P}(S)=\{\varnothing,\{a\},\{b\},\{a,b\}\}$$ with order given by set inclusion $\subseteq$.

• Least element: $\varnothing$
• Greatest element: $\{a,b\}$

So this is a bounded lattice.

Example 2: Divisibility lattice

Consider the set of positive divisors of 12: $$\{1,2,3,4,6,12\}$$ ordered by divisibility.

• Least element: $1$
• Greatest element: $12$

Thus, it is also a bounded lattice.

Hasse diagram for the divisors of 12

      12
     /  \
    4    6
    |   / \
    2  3   |
     \ |  /
       1

This diagram shows the partial order by divisibility, with 1 at the bottom and 12 at the top.

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2. Complement in a Lattice

• A complement of an element $a$ is another element $b$ such that their meet is the least element and their join is the greatest element.
• Intuitively, a complement acts like the “missing part” of an element relative to the whole lattice.

If $b$ is a complement of $a$, then:

• $a \wedge b = 0$ means they have nothing in common at the bottom level,
• $a \vee b = 1$ means together they generate the whole lattice.

Example: Set complement

In the power set lattice $\mathcal{P}(S)$, if $S = \{a,b,c\}$, then for the subset $A=\{a,b\}$, a complement is: $$A^c = \{c\}$$ because:

• $A \cap A^c = \varnothing$,
• $A \cup A^c = S$.

This is the most familiar example of complement in a lattice.

Important observation

In some lattices, complements need not be unique. A lattice may have:

• multiple complements for the same element, or
• no complement at all for certain elements.

So being complemented is a stronger requirement than merely being bounded.

Example of non-uniqueness

In some modular lattices, an element can have more than one complement. This shows that complemented lattices do not automatically behave like set-theoretic complements unless additional properties such as distributivity are present.

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3. Complemented Lattice and Boolean-Like Behavior

• A complemented lattice combines boundedness with the existence of complements for every element.
• This makes the lattice highly structured and especially important in logic and algebra.

A complemented lattice satisfies:

1. It has $0$ and $1$,
2. Every element $a$ has at least one complement $b$.

Why this matters

The presence of complements allows elements to be “split” into parts that together reconstruct the whole lattice. This is similar to:

• true/false in logic,
• a subset and its set-theoretic complement,
• a region and its outside in geometry.

Stronger conditions

If a complemented lattice is also distributive, then:

• complements are unique,
• the lattice becomes a Boolean lattice.

Boolean lattices are the algebraic foundation of:

• propositional logic,
• digital circuit design,
• set operations,
• switching theory.

Example: Boolean lattice of subsets

For $S = \{a,b\}$, the power set lattice is: $$\varnothing,\ \{a\},\ \{b\},\ \{a,b\}$$ Here:

• $\varnothing$ complements $\{a,b\}$,
• $\{a\}$ complements $\{b\}$.

This lattice is complemented and distributive, hence Boolean.

Example of a complemented but non-distributive lattice

A standard example is the diamond lattice $M_3$:

      1
    / | \
   a  b  c
    \ | /
      0

Each of $a,b,c$ is complemented by any of the other two:

• $a$ has complements $b$ and $c$,
• $b$ has complements $a$ and $c$,
• $c$ has complements $a$ and $b$.

So $M_3$ is complemented, but not distributive.

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Working / Process

1. Check whether the lattice is bounded

• Identify the least element $0$ and greatest element $1$.
• In a Hasse diagram, $0$ appears at the bottom and $1$ at the top.
• If either does not exist, the lattice is not bounded.

2. Find meets and joins for elements

• For each pair of elements, determine their meet $\wedge$ and join $\vee$.
• This is essential for checking complements because complements are defined using meet and join.

3. Test for complements

• For a given element $a$, search for an element $b$ such that:
  • $a \wedge b = 0$,
  • $a \vee b = 1$.
• If every element has at least one such $b$, the lattice is complemented.
• If the complement is unique for every element and the lattice is distributive, the lattice is Boolean.

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Advantages / Applications

• Provides a clear algebraic framework for analyzing structure with a smallest and largest element.
• Plays a major role in Boolean algebra, which is fundamental to logic and computer science.
• Helps in understanding set operations, especially union, intersection, and complement.
• Useful in digital circuit design, where logical variables often behave like elements of a Boolean lattice.
• Important in combinatorics and order theory, especially when studying posets and Hasse diagrams.
• Supports algebraic reasoning in modular lattices, distributive lattices, and related structures.
• Helps classify lattices and identify when a lattice behaves like a logical system.

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Summary

• A bounded lattice has a least element $0$ and a greatest element $1$.
• A complemented lattice is a bounded lattice in which every element has a complement.
• Complements satisfy $a \wedge b = 0$ and $a \vee b = 1$.
• If a complemented lattice is distributive, it becomes a Boolean lattice.
• Important terms to remember: bounded lattice, least element, greatest element, complement, meet, join, complemented lattice, Boolean lattice