Boolean algebra homomorphism

Let A and B be Boolean algebrasMathworldPlanetmath. A function f:AB is called a Boolean algebra homomorphism, or homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath for short, if f is a {0,1}-lattice homomorphismMathworldPlanetmath ( such that f respects : f(a)=f(a).

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:

  1. 1.

    if f respects , then f respects iff it respects ;

  2. 2.

    if f is a lattice homomorphism, then f respects 0 and 1 iff it respects .

The first assertion can be shown by de Morgan’s laws. For example, to see the LHS implies RHS, f(ab)=f((ab))=f(ab)=((f(a)f(b))=f(a)f(b)=f(a)′′f(b)′′=f(a)f(b). The second assertion can also be easily proved. For example, to see that the LHS implies RHS, we have that f(a)f(a)=f(aa)=f(1)=1 and f(a)f(a)=f(aa)=f(0)=0. Together, this implies that f(a) is the complementPlanetmathPlanetmath of f(a), which is f(a).

If a function satisfies one, and hence all, of the above conditions also satisfies the property that f(0)=0, for f(0)=f(aa)=f(a)f(a)=f(a)f(a)=0. Dually, f(1)=1.

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 monomorphismMathworldPlanetmathPlanetmath, epimorphismMathworldPlanetmath, endormophism, automorphism, and isomorphismMathworldPlanetmathPlanetmath.

Let f:AB be a Boolean algebra homomorphism. Then the kernel of f is the set {aAf(a)=0}, and is written ker(f). Observe that ker(f) is a Boolean ideal of A.

Let κ be a cardinal. A Boolean algebra homomorphism f:AB is said to be κ-completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if for any subset CA such that

  1. 1.

    |C|κ, and

  2. 2.

    C exists,

then f(C) exists and is equal to f(C). Here, f(C) is the set {f(c)cC}. Note that again, by de Morgan’s laws, if C exists, then f(C) exists and is equal to f(C). If we place no restrictionsPlanetmathPlanetmathPlanetmath on the cardinality of C (i.e., drop condition 1), then f:AB is said to be a complete Boolean algebra homomorphism. In the categoriesMathworldPlanetmath of κ-complete Boolean algebras and complete Boolean algebras, the morphisms are κ-complete homomorphisms and complete homomorphisms respectively.

Title Boolean algebra homomorphism
Canonical name BooleanAlgebraHomomorphism
Date of creation 2013-03-22 18:02:05
Last modified on 2013-03-22 18:02:05
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 06E05
Classification msc 03G05
Classification msc 06B20
Classification msc 03G10
Synonym Boolean homomorphism
Defines kernel
Defines complete Boolean algebra homomorphism
Defines κ-complete Boolean algbra homomorphism