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 homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between algebrasMathworldPlanetmathPlanetmathPlanetmath, a homomorphism Ο•:𝑨→𝑩 between two partial algebrasMathworldPlanetmath of type Ο„ is a function from A to B that satisfies the equation

ϕ⁒(f𝑨⁒(a1,…,an))=f𝑩⁒(ϕ⁒(a1),…,ϕ⁒(an)) (1)

for every n-ary function symbol fβˆˆΟ„. However, because f𝑨 and f𝑩 are not everywhere defined in their respective domains, care must be taken as to what the equation means.

  1. 1.

    Ο• is a homomorphism if, given that f𝑨⁒(a1,…,an) is defined, so is f𝑩⁒(ϕ⁒(a1),…,ϕ⁒(an)), and equation (1) is satisifed.

  2. 2.

    Ο• is a full homomorphism if it is a homomorphism and, given that f𝑩⁒(b1,…,bn) is defined and in ϕ⁒(A), for biβˆˆΟ•β’(A), there exist ai∈A with bi=ϕ⁒(ai), such that f𝑨⁒(a1,…,an) is defined.

  3. 3.

    Ο• is a strong homomorphism if it is a homomorphism and, given that f𝑩⁒(ϕ⁒(a1),…,ϕ⁒(an)) is defined, so is f𝑨⁒(a1,…,an).

We have the following implicationsMathworldPlanetmath:

strong homomorphism β†’ full homomorphism β†’ homomorphism.

For example, field homomorphisms are strong homomorphisms.

Homomorphisms preserve constants: for each constant symbol f in Ο„, ϕ⁒(f𝑨)=f𝑩. In fact, when restricted to constants, Ο• is a bijection between constants of 𝑨 and constants of 𝑩.

When 𝑨 is an algebra (all partial operations are total), a homomorphism from 𝑨 is always strong, so that all three notions of homomorphisms coincide.

An isomorphismMathworldPlanetmathPlanetmath is a bijectiveMathworldPlanetmath homomorphism Ο•:𝑨→𝑩 such that its inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath Ο•-1:𝑩→𝑨 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 subalgebrasMathworldPlanetmath are related. Suppose 𝑨 and 𝑩 are partial algebras of type Ο„. Let Ο•:Aβ†’B be a function, and C=ϕ⁒(A). For each n-ary function symbol fβˆˆΟ„, define n-ary partial operation fπ‘ͺ on C as follows:

for b1,…,bn∈C, fπ‘ͺ⁒(b1,…,bn) is defined iff the set

D:={(a1,…,an)∈Anβˆ£Ο•β’(ai)=bi}∩dom⁑(f𝑨)

is non-empty, where dom⁑(f𝑨) is the domain of definition of f𝑨, and when this is the case, fπ‘ͺ⁒(b1,…,bn):=ϕ⁒(f𝑨⁒(a1,…,an)), for some (a1,…,an)∈D.

If Ο• preserves constants (if any), and fC is non-empty for each fβˆˆΟ„ then π‘ͺ is a partial algebra of type Ο„.

Fix an arbitrary n-ary symbol fβˆˆΟ„. The following are the basic properties of π‘ͺ:

Proposition 1.

Ο• is a homomorphism iff 𝐂 is a weak subalgebra of 𝐁.

Proof.

Suppose first that Ο• is a homomorphism. If n=0, then fπ‘¨βˆˆA, and f𝑩=ϕ⁒(f𝑨)∈C. If n>0, then for some a1,…,an∈A, f𝑨⁒(a1,…,an) is defined, and consequently f𝑩⁒(ϕ⁒(a1),…,ϕ⁒(an)) is defined, and is equal to ϕ⁒(f𝑨⁒(a1,…,an))∈C. By the definition for fπ‘ͺ above, fπ‘ͺ⁒(ϕ⁒(a1),…,ϕ⁒(an)):=ϕ⁒(f𝑨⁒(a1,…,an)). This shows that π‘ͺ is a Ο„-algebra.

To furthermore show that π‘ͺ is a weak subalgebra of 𝑩, assume fπ‘ͺ⁒(b1,…,bn) is defined. Then there are a1,…,an∈A with bi=ϕ⁒(ai) such that f𝑨⁒(a1,…,an) is defined. Since Ο• is a homomorphism, f𝑩⁒(ϕ⁒(a1),…,ϕ⁒(an)), and hence f𝑩⁒(b1,…,bn), is defined. Furthermore, fC⁒(b1,…,bn)=ϕ⁒(f𝑨⁒(a1,…,an))=f𝑩⁒(ϕ⁒(a1),…,ϕ⁒(an))=f𝑩⁒(b1,…,bn). This shows that π‘ͺ is weak.

On the other hand, suppose now that π‘ͺ is a weak subalgebra of 𝑩. Suppose a1,…,an∈A and f𝑨⁒(a1,…,an) is defined. Let bi=ϕ⁒(ai)∈C. Then, by the definition of fπ‘ͺ, fπ‘ͺ⁒(b1,…,bn) is defined and is equal to ϕ⁒(f𝑨⁒(a1,…,an)). Since π‘ͺ is weak, f𝑩⁒(b1,…,bn) is defined and is equal to fπ‘ͺ⁒(b1,…,bn). As a result, ϕ⁒(f𝑨⁒(a1,…,an))=fπ‘ͺ⁒(b1,…,bn)=f𝑩⁒(b1,…,bn)=f𝑩⁒(ϕ⁒(a1),…,ϕ⁒(an)). Hence Ο• is a homomorphism. ∎

Proposition 2.

Ο• is a full homomorphism iff 𝐂 is a relative subalgebra of 𝐁.

Proof.

Suppose first that Ο• is full. Since Ο• is a homomorphism, π‘ͺ is weak. Suppose b1,…,bn∈C such that f𝑨⁒(b1,…,bn) is defined and is in C. Since Ο• is full, there are ai∈A such that bi=ϕ⁒(ai) and f𝑨⁒(a1,…,an) is defined, and ϕ⁒(f𝑨⁒(a1,…,an))=f𝑩⁒(ϕ⁒(a1),…,ϕ⁒(an))=f𝑩⁒(b1,…,bn), so that f𝑩⁒(b1,…,bn) is defined and thus π‘ͺ is a relative subalgebra of 𝑩.

Conversely, suppose that π‘ͺ is a relative subalgebra of 𝑩. Then π‘ͺ is a weak subalgebra of 𝑩 and Ο• is a homomorphism. To show that Ο• is full, suppose that bi∈C such that f𝑩⁒(b1,…,bn) is defined in C. Then fπ‘ͺ⁒(b1,…,bn) is defined in C and is equal to f𝑩⁒(b1,…,bn). This means that there are ai∈A such that bi=ϕ⁒(ai), and f𝑨⁒(a1,…,an) is defined, showing that f𝑨 is full. ∎

Proposition 3.

Ο• is a strong homomorphism iff 𝐂 is a subalgebra of 𝐁.

Proof.

Suppose first that Ο• is strong. Since Ο• is full, π‘ͺ is a relative subalgebra of 𝑩. Suppose now that for bi∈C, f𝑩⁒(b1,…,bn) is defined. Since bi=ϕ⁒(ai) for some ai∈A, and since Ο• is strong, f𝑨⁒(a1,…,an) is defined. This means that f𝑩⁒(b1,…,bn)=f𝑩⁒(ϕ⁒(a1),…,ϕ⁒(an))=ϕ⁒(f𝑨⁒(a1,…,an)), which is in C. So π‘ͺ is a subalgebra of 𝑩.

Going the other direction, suppose now that π‘ͺ is a subalgebra of 𝑩. Since π‘ͺ is a relative subalgebra of 𝑩, Ο• is full. To show that Ο• is strong, suppose f𝑩⁒(ϕ⁒(a1),…,ϕ⁒(an)) is defined. Then fπ‘ͺ⁒(ϕ⁒(a1),…,ϕ⁒(an)) is defined and is equal to f𝑩⁒(ϕ⁒(a1),…,ϕ⁒(an)). By definition of fπ‘ͺ, f𝑨⁒(a1,…,an) is therefore defined. So Ο• is strong. ∎

Definition. Let 𝑨 and 𝑩 be partial algebras of type Ο„. If Ο•:𝑨→𝑩 is a homomorphism, then π‘ͺ, as defined above, is a partial algebra of type Ο„, and is called the homomorphic image of A via Ο•, and is sometimes written ϕ⁒(𝑨).

References

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