0-Bottom (Bottom Element of a Boolean Algebra)

Definition

$$0 \land a = 0 \quad \text{and} \quad 0 \lor a = a$$

In any Boolean algebra (or more generally a bounded lattice), the bottom element is the unique element that is less than or equal to every other element. It is usually denoted $0$ and serves as the identity for the join operation $\lor$ and as an absorbing element for the meet operation $\land$. The existence of a bottom element makes the Boolean algebra a bounded lattice.

Notes

A Boolean algebra is an algebraic structure $(B, \land, \lor, \neg, 0, 1)$ where $\land$ and $\lor$ are binary operations (meet and join), $\neg$ is a unary operation (complement), $0$ is the bottom (or zero) element, and $1$ is the top (or unit) element. The axioms include:

• Idempotence: $a \land a = a$, $a \lor a = a$
• Commutativity: $a \land b = b \land a$, $a \lor b = b \lor a$
• Associativity: $(a \land b) \land c = a \land (b \land c)$, $(a \lor b) \lor c = a \lor (b \lor c)$
• Absorption: $a \land (a \lor b) = a$, $a \lor (a \land b) = a$
• Distributivity: $a \land (b \lor c) = (a \land b) \lor (a \land c)$, $a \lor (b \land c) = (a \lor b) \land (a \lor c)$
• Complements: $a \land \neg a = 0$, $a \lor \neg a = 1$

From these, the properties of the bottom follow:

• $0$ is an identity for $\lor$: $0 \lor a = a$.
• $0$ is an annihilator for $\land$: $0 \land a = 0$.
• The bottom element is unique: if $0$ and $0'$ both satisfy the bottom properties, then $0 = 0 \land 0' = 0'$.

In the standard example of the Boolean algebra of subsets of a set $X$, the bottom element is the empty set $\emptyset$. In propositional logic, interpreted as a Boolean algebra, the bottom is the constant false proposition.

Properties

• Uniqueness: The bottom element is unique in any bounded lattice.
• Complement relation: The complement of the bottom is the top: $\neg 0 = 1$.
• Order-theoretic: In the partial order defined by $a \le b$ iff $a \land b = a$, the bottom is the least element: $0 \le a$ for all $a$.
• Fixed point: For any function $f$ that preserves joins, $f(0) \le 0$ if $f$ is order-preserving; but typically $0$ is not a fixed point.

Examples

1. Subset algebra: For any set $X$, the power set $\mathcal{P}(X)$ with union as $\lor$, intersection as $\land$, and complement as set complement has bottom $\emptyset$.
2. Two-element Boolean algebra: $\{0,1\}$ with the usual logical operations: $0 \land a = 0$, $0 \lor a = a$, $\neg 0 = 1$.
3. Boolean algebra of propositions: The equivalence classes of logical formulas under logical equivalence form a Boolean algebra where the bottom is the class of contradictions (e.g., $p \land \neg p$).

Examples

Given a Boolean algebra $B$, the bottom element $0$ satisfies:

• $0 \land 0 = 0$, $0 \lor 0 = 0$
• $0 \land 1 = 0$, $0 \lor 1 = 1$
• $\neg 0 = 1$

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Significance

Foundation of Boolean Algebra

The bottom element is one of the two distinguished constants (along with the top) that define the structure of a Boolean algebra. It is essential for the definition of complements and for the connection to logic, where false is represented by 0.

Lattice Theory

In the broader context of lattice theory, the existence of a bottom element characterizes bounded lattices. Many results rely on the bottom, such as the Knaster–Tarski fixed-point theorem.

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Glossary

Term / Symbol	Meaning
Boolean algebra	An algebraic structure $(B, \land, \lor, \neg, 0, 1)$ where $\land$ and $\lor$ are meet and join, $\neg$ is complement, $0$ is bottom, $1$ is top, satisfying the axioms of idempotence, commutativity, associativity, absorption, distributivity, and complements.
$0$	The unique bottom element of a Boolean algebra, satisfying $0 \land a = 0$ and $0 \lor a = a$.
$\land$	Meet operation: binary infimum (greatest lower bound).
$\lor$	Join operation: binary supremum (least upper bound).
bounded lattice	A lattice that has both a bottom and a top element.
identity element	An element $e$ such that $e * a = a * e = a$ for all $a$; here $0$ is an identity for $\lor$.
absorbing element	An element $a$ such that $a * b = a$ for all $b$; here $0$ is absorbing for $\land$.