Re: Isomorphic Structures

Quote:

Originally Posted by

**drg** H*Given a structure A and a bijection g with domain |A|, I have to show that there is a unique structure B such that g is an isomorphism of A onto B.*

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..*a structure A* what does that term mean?

Re: Isomorphic Structures

Quote:

Originally Posted by

**drg** *Given a structure A and a bijection g with domain |A| (i.e. the universe of A), I have to show that there is a unique structure B such that g is an isomorphism of A onto B.*

I believe this is a better link that explains what a structure is in the context of mathematical logic.

If I understand correctly, g is an isomorphism between A and B if

(1) g is a bijection between |A| and |B|,

(2) for every functional symbol f and every $\displaystyle \vec{a}\in |A|$ it is the case that $\displaystyle g(f^A(\vec{a}))=f^B(g(\vec{a}))$, and

(3) for every predicate symbol R and every $\displaystyle \vec{a}\in |A|$ it is the case that $\displaystyle R^A(\vec{a})$ iff $\displaystyle R^B(g(\vec{a}))$.

The universe of B must be the image of g. We need to show that there exists one and only one interpretation of functional and predicate symbols on |B| such that g is an isomorphism. Well, this interpretation is uniquely determined by properties (2) and (3).

Re: Isomorphic Structures

Quote:

Originally Posted by

**emakarov** I believe

this is a better link that explains what a structure is in the context of mathematical logic.

If I understand correctly, g is an isomorphism between A and B if

(1) g is a bijection between |A| and |B|,

(2) for every functional symbol f and every $\displaystyle \vec{a}\in |A|$ it is the case that $\displaystyle g(f^A(\vec{a}))=f^B(g(\vec{a}))$, and

(3) for every predicate symbol R and every $\displaystyle \vec{a}\in |A|$ it is the case that $\displaystyle R^A(\vec{a})$ iff $\displaystyle R^B(g(\vec{a}))$.

The universe of B must be the image of g. We need to show that there exists one and only one interpretation of functional and predicate symbols on |B| such that g is an isomorphism. Well, this interpretation is uniquely determined by properties (2) and (3).

Yea my bad that is the article I had in mind.

Thanks that actually helped, even though I was sort of missing the point and realized that the problem basically says: given bijection g there IS a UNIQUE structure B isomorphic to A UNDER g. Might sound redundant but it makes sense to me now.