Hello all(adsbygoogle = window.adsbygoogle || []).push({});

This question relates to products of structures all with the same symbol set [itex]S[/itex]. After I give a little background the question follows.

*Direct Products*

This definition of the direct product is taken from Ebbinghaus, et.al.

Let [itex]I[/itex] be a nonempty set. For every [itex]i\in I[/itex], let [itex]\mathcal{A}_{i}[/itex]

be an [itex]S-[/itex]structure. The domain of the direct product is

[itex]\left\{ g:g\in\left[I\rightarrow\bigcup_{i\in I}A_{i}\right]\wedge\forall i\in I\left(g\left(i\right)\in A_{i}\right)\right\} .[/itex]

Here, [itex]\left[I\rightarrow\bigcup_{i\in I}A_{i}\right][/itex] denotes the

set of all functions whose domain is [itex]I[/itex] and range contained in [itex]\bigcup_{i\in I}A_{i}[/itex].

For [itex]g\in\prod_{i\in I}A_{i}[/itex], we also write [itex]\left\langle g\left(i\right):i\in I\right\rangle [/itex].

For a constant symbol [itex]c[/itex],

[itex]

c^{\mathcal{A}}:=\left\langle c^{\mathcal{A}_{i}}:i\in I\right\rangle .

[/itex]

For an [itex]n-[/itex]ary relation symbol [itex]R[/itex] and for [itex]g_{1},...,g_{n}\in\prod_{i\in I}A_{i}[/itex],

say that [itex]R^{\mathcal{A}}g_{1}...g_{n}[/itex] iff for all [itex]i\in I[/itex], [itex]R^{\mathcal{A}_{i}}g_{1}\left(i\right)...g_{n}(i)[/itex].

For an [itex]n-[/itex]ary function symbol [itex]f[/itex] and for [itex]g_{1},...,g_{n}\in\prod_{i\in I}A_{i}[/itex],

say that

[itex]f^{\mathcal{A}}\left(g_{1},...,g_{n}\right):=\left\langle f^{\mathcal{A}_{i}}\left(g_{1}\left(i\right),...,g_{n}\left(i\right)\right):i\in I\right\rangle .[/itex]

*Partial Isomorphisms (One-to-one Homomorphisms)*

Ebbinghaus defines a partial isomorphism to be an injective homomorphism

on page 180.

Suppose [itex]\mathcal{A}[/itex] and [itex]\mathcal{B}[/itex] are [itex]S-[/itex]structures and

[itex]p[/itex] is a map whose domain is a subset of [itex]A[/itex] and range is a subset

of [itex]B[/itex]. Then [itex]p[/itex] is called a partial isomorphism if

[itex]p[/itex] is injective

[itex]p[/itex] is a homomorphism in the following sense

for any constant symbol [itex]c[/itex] and any [itex]a\in\mathsf{dom}\left(p\right)[/itex],

[itex]c^{\mathcal{A}}=a[/itex] iff [itex]c^{\mathcal{B}}=p\left(a\right)[/itex]

for any [itex]n-[/itex]ary relation symbol [itex]R[/itex] and [itex]a_{1},...,a_{n}\in\mathsf{dom}\left(p\right)[/itex],

[itex]R^{\mathcal{A}}a_{1}...a_{n}[/itex] iff [itex]R^{\mathcal{B}}p(a_{1})...p(a_{n})[/itex]

for any [itex]n-[/itex]ary function symbol [itex]f[/itex] and [itex]a_{1},...,a_{n}\in\mathsf{dom}\left(p\right)[/itex],

[itex]f^{\mathcal{A}}(a_{1},...,a_{n})=a[/itex] iff [itex]f^{\mathcal{B}}(p(a_{1}),...,p(a_{n}))=p(a)[/itex]

**The Question**

Do there exist maps [itex]p_{i}:A_{i}\rightarrow A[/itex] that are partial isomorphisms?

If not, then is there any way to take a family of structures and "create" a new structure that each structure in the family can be homomorphically injected into the new structure?

This is an answer someone gave me:

More generally, if you are working in a category of algebras where:

1. For any two objects [itex]A[/itex] and [itex]B[/itex], there is always at least one morphism from [itex]A[/itex] to [itex]B[/itex]; and

2. For any family [itex]\{A_i\}_{i\in I}[/itex] of objects there is a (categorical) product,

then the product of the family of structures will always work (assuming the Axiom of Choice): for given a family [itex]\{A_i\}_{i\in I}[/itex], if [itex](P,\pi_i)[/itex] is the product, then for pair [itex](i,j)[/itex] let [itex]f_{ij}\colon A_i\to A_j[/itex] be an arbitrary morphism if [itex]i\neq j[/itex], and let [itex]f_{ij}=\mathrm{id}_{A_i}[/itex] if [itex]i=j[/itex]. Then the family [itex]f_{i_0,j}\colon A_{i_0}\to A_j[/itex] induces a homomoprhism into the product [itex]\mathcal{F}_{i_0}\colon A_{i_0}\to P[/itex] such that [itex]f_{i_0,j}=\pi_j\circ \mathcal{F}_{i_0}[/itex] for each [itex]j[/itex]; in particular, [itex]\mathrm{id}_{A_{i_0}} = \pi_{i_0}\circ \mathcal{F}_{i_0}[/itex], so [itex]\mathcal{F}_{i_0}[/itex] must be one-to-one, giving the desired immersion.How do we know that[itex]\mathcal{F}_{i_0}[/itex]homomorphismexists?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Injective homomorphism into an amalgam of structrues

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**