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When we study Algebra we find the concepts of Homomorphism and Isomorphism all over the place. When two structures (rings, fields, vector spaces, ...) are Isomorphic we say (quite informally) that they are "algebraically indistinguishable", or "algebraically the same".

My question is the following: How do I know if a property is "algebraic"? What are the properties/constructions of $\mathbb{R}^2$ that are preserved by vector spaces isomorphisms?

I mean, the real vector spaces $\mathbb{R}^2$ and $\mathscr{P}_1(\mathbb{R})$ are isomorphic, but each of the sets $\mathbb{R}^2$ and $\mathscr{P}_1(\mathbb{R})$ have more structure than that of a vector space. How do I know if a construction in $\mathbb{R}^2$ can be carried over to a analogous construction in $\mathscr{P}_1(\mathbb{R})$? More than that, why I can't (can I?) use "transport of structure" to define a product in $\mathbb{R}^2$ using the product of $\mathscr{P}_1(\mathbb{R})$?

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  • $\begingroup$ The product of polynomials of the same degree doubles the degree. You can define a function $\mathbb R^2\times \mathbb R^2 \to \mathbb R^3$ that mirrors polynomial multiplication: $((a,b),(c,d)) \mapsto (ac, ad+bc, bd)$ $\endgroup$ – David Peterson Sep 16 '16 at 22:48
  • $\begingroup$ What's $\mathscr{P}_1(\mathbb{R})$? $\endgroup$ – Rob Arthan Sep 17 '16 at 0:09
  • $\begingroup$ @RobArthan Polynomials with degree $\leq$ 1 $\endgroup$ – Pugglo Sep 17 '16 at 1:54
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    $\begingroup$ @Puggio Rob asks because it's not standard notation, so you should probably define it in the post. $\endgroup$ – Matt Samuel Sep 17 '16 at 2:50
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Fundamentally, in any of these settings (groups, vector spaces, etc.) there is an underlying language. The language usually consists of some

  • underlying sets (in the case of groups, rings, and fields, just one set; in the case of vector spaces, there a field $F$ and a vector space $X$, which is two sets)

  • functions on the sets ($*$ in the case of groups, $\cdot$ and $+$ in the case of rings and the case of vector spaces)

  • constants ($e$ in the case of groups, $0$ and $1$ in the case of rings and fields, $

  • relations ($<$, in the case of ordered fields).

From the language, we can build up a set of mathematical statements, or sentences, that are meaningful things to say. Technically speaking, we should allow at a minimum all second-order sentences (not just first-order!).

An algebraic property in this sense (though I would prefer a term like definable property) is one that can be expressed by a sentence in the language.

Some notes:

  • The key idea here is that of a language. When we are talking about groups, it does not make sense to say that $xy = x + y$ for some $x,y$. That is because groups do not have a meaning for the $+$ function. But it can be more subtle. When we are talking about vector spaces, we are not allowed to use the norm, $\| \cdot \|$ in our language. In fact, any property involving the norm is no longer preserved under vector space isomorphism. Although it would be preserved under an isomorphism of normed vector spaces.

  • Second-order sentences are, very roughly, everything you can say with quantifiers, and, not, equals, and so on. They're called second-order because you're allowed to quantify over subsets or functions as well as elements of the group, ring, vector space, or other object in question.

  • We may want even more than all second-order sentences. One idea would be to say that an infinite conjunction and infinite disjunction of algebraic properties is algebraic. Quantification is basically a special case of this, anyway. This will give us a very large class of sentences, indeed.

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