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:= 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 :
Proposition 1.
is a homomorphism iff is a weak subalgebra of .
Proof.
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. β
Proposition 2.
is a full homomorphism iff is a relative subalgebra of .
Proof.
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. β
Proposition 3.
is a strong homomorphism iff is a subalgebra of .
Proof.
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. 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
-
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 |