homomorphism between partial algebras
Like subalgebras of partial algebras, there are also three ways to define homomorphisms between partial algebras. Similar to the definition of homomorphisms between algebras, a homomorphism between two partial algebras of type is a function from to that satisfies the equation
for every -ary function symbol . However, because and are not everywhere defined in their respective domains, care must be taken as to what the equation means.
is a homomorphism if, given that is defined, so is , and equation (1) is satisifed.
is a full homomorphism if it is a homomorphism and, given that is defined and in , for , there exist with , such that is defined.
is a strong homomorphism if it is a homomorphism and, given that is defined, so is .
We have the following implications:
strong homomorphism full homomorphism homomorphism.
For example, field homomorphisms are strong homomorphisms.
When is an algebra (all partial operations are total), a homomorphism from is always strong, so that all three notions of homomorphisms coincide.
The various types of homomorphisms and the various types of subalgebras are related. Suppose and are partial algebras of type . Let be a function, and . For each -ary function symbol , define -ary partial operation on as follows:
for , is defined iff the set
is non-empty, where is the domain of definition of , and when this is the case, , for some .
If preserves constants (if any), and is non-empty for each then is a partial algebra of type .
Fix an arbitrary -ary symbol . The following are the basic properties of :
is a homomorphism iff is a weak subalgebra of .
Suppose first that is a homomorphism. If , then , and . If , then for some , is defined, and consequently is defined, and is equal to . By the definition for above, . This shows that is a -algebra.
To furthermore show that is a weak subalgebra of , assume is defined. Then there are with such that is defined. Since is a homomorphism, , and hence , is defined. Furthermore, . This shows that is weak.
On the other hand, suppose now that is a weak subalgebra of . Suppose and is defined. Let . Then, by the definition of , is defined and is equal to . Since is weak, is defined and is equal to . As a result, . Hence is a homomorphism. ∎
is a full homomorphism iff is a relative subalgebra of .
Suppose first that is full. Since is a homomorphism, is weak. Suppose such that is defined and is in . Since is full, there are such that and is defined, and , so that is defined and thus is a relative subalgebra of .
Conversely, suppose that is a relative subalgebra of . Then is a weak subalgebra of and is a homomorphism. To show that is full, suppose that such that is defined in . Then is defined in and is equal to . This means that there are such that , and is defined, showing that is full. ∎
is a strong homomorphism iff is a subalgebra of .
Suppose first that is strong. Since is full, is a relative subalgebra of . Suppose now that for , is defined. Since for some , and since is strong, is defined. This means that , which is in . So is a subalgebra of .
Going the other direction, suppose now that is a subalgebra of . Since is a relative subalgebra of , is full. To show that is strong, suppose is defined. Then is defined and is equal to . By definition of , is therefore defined. So is strong. ∎
Definition. Let and be partial algebras of type . If is a homomorphism, then , as defined above, is a partial algebra of type , and is called the homomorphic image of via , and is sometimes written .
- 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
|Title||homomorphism between partial algebras|
|Date of creation||2013-03-22 18:42:57|
|Last modified on||2013-03-22 18:42:57|
|Last modified by||CWoo (3771)|