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Injective homomorphism into an amalgam of structrues

  1. Feb 6, 2012 #1
    Hello all

    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:
    How do we know that homomorphism [itex]\mathcal{F}_{i_0}[/itex] exists?
     
  2. jcsd
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