PlanetMath: Boolean algebra homomorphism

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Boolean algebra homomorphism

(Definition)

Let and be Boolean algebras. A function $f:A\to B$ is called a Boolean algebra homomorphism, or homomorphism for short, if is a $\lbrace 0,1\rbrace$ -lattice homomorphism such that respects : .

Typically, to show that a function between two Boolean algebras is a Boolean algebra homomorphism, it is not necessary to check every defining condition. In fact, we have the following:

if respects , then respects $\vee$ iff it respects $\wedge$ ;
if is a lattice homomorphism, then respects 0 and iff it respects .

The first assertion can be shown by de Morgan's laws. For example, to see the LHS implies RHS, $f(a\wedge b)= f((a'\vee b')')= f(a'\vee b')'=((f(a')\vee f(b'))'= f(a')'\wedge f(b')'= f(a)'' \wedge f(b)'' = f(a)\wedge f(b)$ . The second assertion can also be easily proved. For example, to see that the LHS implies RHS, we have that $f(a')\vee f(a)= f(a'\vee a)=f(1)=1$ and $f(a')\wedge f(a)=f(a'\wedge a)=f(0)=0$ . Together, this implies that

is the complement of

, which is

If a function satisfies one, and hence all, of the above conditions also satisfies the property that , for $f(0)=f(a\wedge a')=f(a)\wedge f(a')= f(a)\wedge f(a)'=0$ . Dually, .

As a Boolean algebra is an algebraic system, the definition of a Boolean algebra homormphism is just a special case of an algebra homomorphism between two algebraic systems. Therefore, one may similarly define a Boolean algebra monomorphism, epimorphism, endormophism, automorphism, and isomorphism.

Let $f:A\to B$ be a Boolean algebra homomorphism. Then the kernel of is the set $\lbrace a\in A\mid f(a)=0\rbrace$ , and is written $\ker(f)$ . Observe that $\ker(f)$ is a Boolean ideal of .

Let $\kappa$ be a cardinal. A Boolean algebra homomorphism $f:A\to B$ is said to be $\kappa$ -complete if for any subset $C\subseteq A$ such that

$\vert C\vert\le \kappa$ , and
$\bigvee C$ exists,

then $\bigvee f(C)$ exists and is equal to $f(\bigvee C)$ . Here,

is the set $\lbrace f(c)\mid c\in C\rbrace$ . Note that again, by de Morgan's laws, if $\bigwedge C$ exists, then $\bigwedge f(C)$ exists and is equal to $f(\bigwedge C)$ . If we place no restrictions on the cardinality of

(i.e., drop condition 1), then $f:A\to B$ is said to be a complete Boolean algebra homomorphism. In the categories of $\kappa$ -complete Boolean algebras and complete Boolean algebras, the morphisms are $\kappa$ -complete homomorphisms and complete homomorphisms respectively.

"Boolean algebra homomorphism" is owned by CWoo.

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Other names:

Boolean homomorphism

Also defines:

kernel, complete Boolean algebra homomorphism, $\kappa$ -complete Boolean algbra homomorphism

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Cross-references: complete, complete Boolean algebras, categories, cardinality, restrictions, place, subset, cardinal, Boolean ideal, algebra, algebraic system, property, satisfies, complement, implies, de Morgan's laws, lattice homomorphism, iff, necessary, homomorphism, function, Boolean algebras
There are 11 references to this entry.

This is version 2 of Boolean algebra homomorphism, born on 2008-04-29, modified 2008-04-29.
Object id is 10554, canonical name is BooleanAlgebraHomomorphism.
Accessed 183 times total.

Classification:

AMS MSC:	03G10 (Mathematical logic and foundations :: Algebraic logic :: Lattices and related structures)
	06B20 (Order, lattices, ordered algebraic structures :: Lattices :: Varieties of lattices)
	03G05 (Mathematical logic and foundations :: Algebraic logic :: Boolean algebras)
	06E05 (Order, lattices, ordered algebraic structures :: Boolean algebras :: Structure theory)

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