homomorphism between partial algebras
Definition
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 Ο:π¨βπ© between two partial algebras
of type Ο is a function from A to B that satisfies the equation
Ο(fπ¨(a1,β¦,an))=fπ©(Ο(a1),β¦,Ο(an)) | (1) |
for every n-ary function symbol fβΟ. However, because fπ¨ and fπ© are not everywhere defined in their respective domains, care must be taken as to what the equation means.
-
1.
Ο is a homomorphism if, given that fπ¨(a1,β¦,an) is defined, so is fπ©(Ο(a1),β¦,Ο(an)), and equation (1) is satisifed.
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2.
Ο is a full homomorphism if it is a homomorphism and, given that fπ©(b1,β¦,bn) is defined and in Ο(A), for biβΟ(A), there exist aiβA with bi=Ο(ai), such that fπ¨(a1,β¦,an) is defined.
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3.
Ο is a strong homomorphism if it is a homomorphism and, given that fπ©(Ο(a1),β¦,Ο(an)) is defined, so is fπ¨(a1,β¦,an).
We have the following implications:
strong homomorphism β full homomorphism β homomorphism.
For example, field homomorphisms are strong homomorphisms.
Homomorphisms preserve constants: for each constant symbol f in Ο, Ο(fπ¨)=fπ©. In fact, when restricted to constants, Ο is a bijection between constants of π¨ and constants of π©.
When π¨ is an algebra (all partial operations are total), a homomorphism from π¨ is always strong, so that all three notions of homomorphisms coincide.
An isomorphism is a bijective
homomorphism Ο:π¨βπ© such that its inverse
Ο-1:π©βπ¨ is also a homomorphism. An embedding is an injective homomorphism. Isomorphisms and full embeddings are strong.
Homomorphic Images
The various types of homomorphisms and the various types of subalgebras are related. Suppose π¨ and π© are partial algebras of type Ο. Let Ο:AβB be a function, and C=Ο(A). For each n-ary function symbol fβΟ, define n-ary partial operation fπͺ on C as follows:
for b1,β¦,bnβC, fπͺ(b1,β¦,bn) is defined iff the set
D:={(a1,β¦,an)βAnβ£Ο(ai)=bi}β©dom(fπ¨) is non-empty, where dom(fπ¨) is the domain of definition of fπ¨, and when this is the case, fπͺ(b1,β¦,bn):=Ο(fπ¨(a1,β¦,an)), for some (a1,β¦,an)βD.
If Ο preserves constants (if any), and fC is non-empty for each fβΟ then πͺ is a partial algebra of type Ο.
Fix an arbitrary n-ary symbol fβΟ. The following are the basic properties of πͺ:
Proposition 1.
Ο is a homomorphism iff π is a weak subalgebra of π.
Proof.
Suppose first that Ο is a homomorphism. If n=0, then fπ¨βA, and fπ©=Ο(fπ¨)βC. If n>0, then for some a1,β¦,anβA, fπ¨(a1,β¦,an) is defined, and consequently fπ©(Ο(a1),β¦,Ο(an)) is defined, and is equal to Ο(fπ¨(a1,β¦,an))βC. By the definition for fπͺ above, fπͺ(Ο(a1),β¦,Ο(an)):=Ο(fπ¨(a1,β¦,an)). This shows that πͺ is a Ο-algebra.
To furthermore show that πͺ is a weak subalgebra of π©, assume fπͺ(b1,β¦,bn) is defined. Then there are a1,β¦,anβA with bi=Ο(ai) such that fπ¨(a1,β¦,an) is defined. Since Ο is a homomorphism, fπ©(Ο(a1),β¦,Ο(an)), and hence fπ©(b1,β¦,bn), is defined. Furthermore, fC(b1,β¦,bn)=Ο(fπ¨(a1,β¦,an))=fπ©(Ο(a1),β¦,Ο(an))=fπ©(b1,β¦,bn). This shows that πͺ is weak.
On the other hand, suppose now that πͺ is a weak subalgebra of π©. Suppose a1,β¦,anβA and fπ¨(a1,β¦,an) is defined. Let bi=Ο(ai)βC. Then, by the definition of fπͺ, fπͺ(b1,β¦,bn) is defined and is equal to Ο(fπ¨(a1,β¦,an)). Since πͺ is weak, fπ©(b1,β¦,bn) is defined and is equal to fπͺ(b1,β¦,bn). As a result, Ο(fπ¨(a1,β¦,an))=fπͺ(b1,β¦,bn)=fπ©(b1,β¦,bn)=fπ©(Ο(a1),β¦,Ο(an)). Hence Ο is a homomorphism. β
Proposition 2.
Ο is a full homomorphism iff π is a relative subalgebra of π.
Proof.
Suppose first that Ο is full. Since Ο is a homomorphism, πͺ is weak. Suppose b1,β¦,bnβC such that fπ¨(b1,β¦,bn) is defined and is in C. Since Ο is full, there are aiβA such that bi=Ο(ai) and fπ¨(a1,β¦,an) is defined, and Ο(fπ¨(a1,β¦,an))=fπ©(Ο(a1),β¦,Ο(an))=fπ©(b1,β¦,bn), so that fπ©(b1,β¦,bn) is defined and thus πͺ is a relative subalgebra of π©.
Conversely, suppose that πͺ is a relative subalgebra of π©. Then πͺ is a weak subalgebra of π© and Ο is a homomorphism. To show that Ο is full, suppose that biβC such that fπ©(b1,β¦,bn) is defined in C. Then fπͺ(b1,β¦,bn) is defined in C and is equal to fπ©(b1,β¦,bn). This means that there are aiβA such that bi=Ο(ai), and fπ¨(a1,β¦,an) is defined, showing that fπ¨ is full. β
Proposition 3.
Ο is a strong homomorphism iff π is a subalgebra of π.
Proof.
Suppose first that Ο is strong. Since Ο is full, πͺ is a relative subalgebra of π©. Suppose now that for biβC, fπ©(b1,β¦,bn) 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. β
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 .
References
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1
G. GrΓ€tzer: Universal Algebra
, 2nd Edition, Springer, New York (1978).
Title | homomorphism between partial algebras |
Canonical name | HomomorphismBetweenPartialAlgebras |
Date of creation | 2013-03-22 18:42:57 |
Last modified on | 2013-03-22 18:42:57 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 13 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 08A55 |
Classification | msc 03E99 |
Classification | msc 08A62 |
Defines | homomorphism |
Defines | full homomorphism |
Defines | strong homomorphism |
Defines | isomorphism |
Defines | strong |
Defines | homomorphic image |
Defines | embedding |