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# 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 $\phi:\boldsymbol{A}\to\boldsymbol{B}$ between two partial algebras of type $\tau$ is a function from $A$ to $B$ that satisfies the equation

$\phi(f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n}))=f_{{\boldsymbol{B}}}(\phi(a_{1}% ),\ldots,\phi(a_{n}))$ | (1) |

for every $n$-ary function symbol $f\in\tau$. However, because $f_{{\boldsymbol{A}}}$ and $f_{{\boldsymbol{B}}}$ are not everywhere defined in their respective domains, care must be taken as to what the equation means.

1. $\phi$ is a

*homomorphism*if, given that $f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n})$ is defined, so is $f_{{\boldsymbol{B}}}(\phi(a_{1}),\ldots,\phi(a_{n}))$, and equation (1) is satisifed.2. $\phi$ is a

*full homomorphism*if it is a homomorphism and, given that $f_{{\boldsymbol{B}}}(b_{1},\ldots,b_{n})$ is defined and in $\phi(A)$, for $b_{i}\in\phi(A)$, there exist $a_{i}\in A$ with $b_{i}=\phi(a_{i})$, such that $f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n})$ is defined.3. $\phi$ is a

*strong homomorphism*if it is a homomorphism and, given that $f_{{\boldsymbol{B}}}(\phi(a_{1}),\ldots,\phi(a_{n}))$ is defined, so is $f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n})$.

We have the following implications:

strong homomorphism $\rightarrow$ full homomorphism $\rightarrow$ homomorphism.

For example, field homomorphisms are strong homomorphisms.

Homomorphisms preserve constants: for each constant symbol $f$ in $\tau$, $\phi(f_{{\boldsymbol{A}}})=f_{{\boldsymbol{B}}}$. In fact, when restricted to constants, $\phi$ is a bijection between constants of $\boldsymbol{A}$ and constants of $\boldsymbol{B}$.

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

An *isomorphism* is a bijective homomorphism $\phi:\boldsymbol{A}\to\boldsymbol{B}$ such that its inverse $\phi^{{-1}}:\boldsymbol{B}\to\boldsymbol{A}$ 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 subalgebras are related. Suppose $\boldsymbol{A}$ and $\boldsymbol{B}$ are partial algebras of type $\tau$. Let $\phi:A\to B$ be a function, and $C=\phi(A)$. For each $n$-ary function symbol $f\in\tau$, define $n$-ary partial operation $f_{{\boldsymbol{C}}}$ on $C$ as follows:

for $b_{1},\ldots,b_{n}\in C$, $f_{{\boldsymbol{C}}}(b_{1},\ldots,b_{n})$ is defined iff the set

$D:=\{(a_{1},\ldots,a_{n})\in A^{n}\mid\phi(a_{i})=b_{i}\}\cap\operatorname{dom% }(f_{{\boldsymbol{A}}})$ is non-empty, where $\operatorname{dom}(f_{{\boldsymbol{A}}})$ is the domain of definition of $f_{{\boldsymbol{A}}}$, and when this is the case, $f_{{\boldsymbol{C}}}(b_{1},\ldots,b_{n}):=\phi(f_{{\boldsymbol{A}}}(a_{1},% \ldots,a_{n}))$, for some $(a_{1},\ldots,a_{n})\in D$.

If $\phi$ preserves constants (if any), and $f_{C}$ is non-empty for each $f\in\tau$ then $\boldsymbol{C}$ is a partial algebra of type $\tau$.

Fix an arbitrary $n$-ary symbol $f\in\tau$. The following are the basic properties of $\boldsymbol{C}$:

###### Proposition 1.

$\phi$ is a homomorphism iff $\boldsymbol{C}$ is a weak subalgebra of $\boldsymbol{B}$.

###### Proof.

Suppose first that $\phi$ is a homomorphism. If $n=0$, then $f_{{\boldsymbol{A}}}\in A$, and $f_{{\boldsymbol{B}}}=\phi(f_{{\boldsymbol{A}}})\in C$. If $n>0$, then for some $a_{1},\ldots,a_{n}\in A$, $f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n})$ is defined, and consequently $f_{{\boldsymbol{B}}}(\phi(a_{1}),\ldots,\phi(a_{n}))$ is defined, and is equal to $\phi(f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n}))\in C$. By the definition for $f_{{\boldsymbol{C}}}$ above, $f_{{\boldsymbol{C}}}(\phi(a_{1}),\ldots,\phi(a_{n})):=\phi(f_{{\boldsymbol{A}}% }(a_{1},\ldots,a_{n}))$. This shows that $\boldsymbol{C}$ is a $\tau$-algebra.

To furthermore show that $\boldsymbol{C}$ is a weak subalgebra of $\boldsymbol{B}$, assume $f_{{\boldsymbol{C}}}(b_{1},\ldots,b_{n})$ is defined. Then there are $a_{1},\ldots,a_{n}\in A$ with $b_{i}=\phi(a_{i})$ such that $f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n})$ is defined. Since $\phi$ is a homomorphism, $f_{{\boldsymbol{B}}}(\phi(a_{1}),\ldots,\phi(a_{n}))$, and hence $f_{{\boldsymbol{B}}}(b_{1},\ldots,b_{n})$, is defined. Furthermore, $f_{C}(b_{1},\ldots,b_{n})=\phi(f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n}))=f_{{% \boldsymbol{B}}}(\phi(a_{1}),\ldots,\phi(a_{n}))=f_{{\boldsymbol{B}}}(b_{1},% \ldots,b_{n})$. This shows that $\boldsymbol{C}$ is weak.

On the other hand, suppose now that $\boldsymbol{C}$ is a weak subalgebra of $\boldsymbol{B}$. Suppose $a_{1},\ldots,a_{n}\in A$ and $f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n})$ is defined. Let $b_{i}=\phi(a_{i})\in C$. Then, by the definition of $f_{{\boldsymbol{C}}}$, $f_{{\boldsymbol{C}}}(b_{1},\ldots,b_{n})$ is defined and is equal to $\phi(f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n}))$. Since $\boldsymbol{C}$ is weak, $f_{{\boldsymbol{B}}}(b_{1},\ldots,b_{n})$ is defined and is equal to $f_{{\boldsymbol{C}}}(b_{1},\ldots,b_{n})$. As a result, $\phi(f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n}))=f_{{\boldsymbol{C}}}(b_{1},% \ldots,b_{n})=f_{{\boldsymbol{B}}}(b_{1},\ldots,b_{n})=f_{{\boldsymbol{B}}}(% \phi(a_{1}),\ldots,\phi(a_{n}))$. Hence $\phi$ is a homomorphism. ∎

###### Proposition 2.

$\phi$ is a full homomorphism iff $\boldsymbol{C}$ is a relative subalgebra of $\boldsymbol{B}$.

###### Proof.

Suppose first that $\phi$ is full. Since $\phi$ is a homomorphism, $\boldsymbol{C}$ is weak. Suppose $b_{1},\ldots,b_{n}\in C$ such that $f_{{\boldsymbol{A}}}(b_{1},\ldots,b_{n})$ is defined and is in $C$. Since $\phi$ is full, there are $a_{i}\in A$ such that $b_{i}=\phi(a_{i})$ and $f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n})$ is defined, and $\phi(f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n}))=f_{{\boldsymbol{B}}}(\phi(a_{1}% ),\ldots,\phi(a_{n}))=f_{{\boldsymbol{B}}}(b_{1},\ldots,b_{n})$, so that $f_{{\boldsymbol{B}}}(b_{1},\ldots,b_{n})$ is defined and thus $\boldsymbol{C}$ is a relative subalgebra of $\boldsymbol{B}$.

Conversely, suppose that $\boldsymbol{C}$ is a relative subalgebra of $\boldsymbol{B}$. Then $\boldsymbol{C}$ is a weak subalgebra of $\boldsymbol{B}$ and $\phi$ is a homomorphism. To show that $\phi$ is full, suppose that $b_{i}\in C$ such that $f_{{\boldsymbol{B}}}(b_{1},\ldots,b_{n})$ is defined in $C$. Then $f_{{\boldsymbol{C}}}(b_{1},\ldots,b_{n})$ is defined in $C$ and is equal to $f_{{\boldsymbol{B}}}(b_{1},\ldots,b_{n})$. This means that there are $a_{i}\in A$ such that $b_{i}=\phi(a_{i})$, and $f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n})$ is defined, showing that $f_{{\boldsymbol{A}}}$ is full. ∎

###### Proposition 3.

$\phi$ is a strong homomorphism iff $\boldsymbol{C}$ is a subalgebra of $\boldsymbol{B}$.

###### Proof.

Suppose first that $\phi$ is strong. Since $\phi$ is full, $\boldsymbol{C}$ is a relative subalgebra of $\boldsymbol{B}$. Suppose now that for $b_{i}\in C$, $f_{{\boldsymbol{B}}}(b_{1},\ldots,b_{n})$ is defined. Since $b_{i}=\phi(a_{i})$ for some $a_{i}\in A$, and since $\phi$ is strong, $f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n})$ is defined. This means that $f_{{\boldsymbol{B}}}(b_{1},\ldots,b_{n})=f_{{\boldsymbol{B}}}(\phi(a_{1}),% \ldots,\phi(a_{n}))=\phi(f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n}))$, which is in $C$. So $\boldsymbol{C}$ is a subalgebra of $\boldsymbol{B}$.

Going the other direction, suppose now that $\boldsymbol{C}$ is a subalgebra of $\boldsymbol{B}$. Since $\boldsymbol{C}$ is a relative subalgebra of $\boldsymbol{B}$, $\phi$ is full. To show that $\phi$ is strong, suppose $f_{{\boldsymbol{B}}}(\phi(a_{1}),\ldots,\phi(a_{n}))$ is defined. Then $f_{{\boldsymbol{C}}}(\phi(a_{1}),\ldots,\phi(a_{n}))$ is defined and is equal to $f_{{\boldsymbol{B}}}(\phi(a_{1}),\ldots,\phi(a_{n}))$. By definition of $f_{{\boldsymbol{C}}}$, $f_{{\boldsymbol{A}}}(a_{1},\ldots,a_{n})$ is therefore defined. So $\phi$ is strong. ∎

Definition. Let $\boldsymbol{A}$ and $\boldsymbol{B}$ be partial algebras of type $\tau$. If $\phi:\boldsymbol{A}\to\boldsymbol{B}$ is a homomorphism, then $\boldsymbol{C}$, as defined above, is a partial algebra of type $\tau$, and is called the *homomorphic image* of $A$ via $\phi$, and is sometimes written $\phi(\boldsymbol{A})$.

# References

- 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).

## Mathematics Subject Classification

08A55*no label found*03E99

*no label found*08A62

*no label found*

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