Boolean algebra homomorphism
Let and be Boolean algebras. A function is called a Boolean algebra homomorphism, or homomorphism for short, if is a -lattice homomorphism (http://planetmath.org/LatticeHomomorphism) 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 iff it respects ;
if is a lattice homomorphism, then respects and iff it respects .
The first assertion can be shown by de Morgan’s laws. For example, to see the LHS implies RHS, . The second assertion can also be easily proved. For example, to see that the LHS implies RHS, we have that and . 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 . 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 be a Boolean algebra homomorphism. Then the kernel of is the set , and is written . Observe that is a Boolean ideal of .
then exists and is equal to . Here, is the set . Note that again, by de Morgan’s laws, if exists, then exists and is equal to . If we place no restrictions on the cardinality of (i.e., drop condition 1), then is said to be a complete Boolean algebra homomorphism. In the categories of -complete Boolean algebras and complete Boolean algebras, the morphisms are -complete homomorphisms and complete homomorphisms respectively.
|Title||Boolean algebra homomorphism|
|Date of creation||2013-03-22 18:02:05|
|Last modified on||2013-03-22 18:02:05|
|Last modified by||CWoo (3771)|
|Defines||complete Boolean algebra homomorphism|
|Defines||-complete Boolean algbra homomorphism|