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Preamble: My previous education was focused either on classical analysis (which was given in quite old traditions, I guess) or on applied Mathematics. Since I was feeling lack of knowledge in 'modern' maths, I have some time now while doing my PhD to learn things I'm interested in by myself.

My impression is that in the last one-two centuries mathematicians put much effort to categorize their knowledge which led finally to abstract algebra and category theory. I didn't learn deeply non of these subjects so I will understand if your answer/comment will be a link to wikipedia page about category theory. I've already read it and it does not answer question. This is not a lazy interest, it is quite important for my understanding of things.

Question Description: My impression is that there are four clearly distinguishable types of mathematical structures on sets, i.e., ways of thinking of just a collection of elements as something meaningful:

  1. Set-theoretical: relations (order, equivalence, etc.)

  2. Algebraic: groups, algebras, fields, vector spaces etc.

  3. Geometrical: topology, metric, smooth structure etc.

  4. Measure-theoretical: $\sigma$-algebras, independence etc.

Some structure could be combined leading to e.g. $(1,2)$ - cosets, $(1,3)$ - quotient topology, $(2,3)$ - topological groups, $(2,4)$ - Haar measure, $(3,4)$ - Hausdorff measure etc.

I guess that any of these structures can be be restated just a relation on some set, but non-abstracted way of thinking of them is more convenient.

Questions:

  1. if my impression is right?

  2. are there any other structures? e.g. I'm interested if there are any dynamical structures corresponding to directed graphs, Markov Chains and other dynamical systems. These objects endow state space with notions of transitivity, absorbance, recurrence equilibrium etc.

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    Concerning the order topology: On a finite set, a topology really is *the same* as a preorder. So here, (1) and (3) are not combined but turn out to yield the same thing.2011-11-03
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    @Rasmus I see, changed it to quotient topology. I also failed to find a nice example for the combination $(1,4)$2011-11-03
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    @Gortaur: [ultrafilters on a set can be seen as finitely additive measures](http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/) on the power set. They would tie in nicely with 2 and 3 as well via [amenable groups](http://en.wikipedia.org/wiki/Amenable_group) and [Banach-Tarski](http://en.wikipedia.org/wiki/Banach-Tarski_paradox).2011-11-03
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    @t.b. thanks, I'll take a look2011-11-03
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    I threw in three additional tags as I don't like questions only tagged with (soft-question). But unfortunately they may not be 100% appropriate; they are the best I can find.2011-11-03
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    @Willie: thanks a lot! I was thinking of other tags - but didn't know about [intuition] and [meta-math] while having doubts about [category-theory]2011-11-03
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    Reminds me of beginning of http://terrytao.wordpress.com/2009/10/19. I think the characterization "four clearly distinguishable types" is contestable, for example a Boolean algebra can be seen as a partially ordered set (1) or algebra (2), but the definitions do not use two fields at once, unlike other examples (topological group, Haar measure, Hausdorff measure...).2011-11-03
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    There are partial algebraic structures, essentially algebraic theories http://ncatlab.org/nlab/show/essentially+algebraic+theory. Sorry, I do not have a good reference.2011-11-10
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    Nice question! I wonder what "independence" as a kind of structure is defined? I cannot find it in Wikipedia. Thanks!2012-03-28
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    @Tim: I meant an independence in the sense of probability theory: $\mathsf P(A\cap B) = \mathsf P(A)\mathsf P(B)$; as for my I used to consider it as an additional structure on the measure space2012-03-29
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    @Ilya: Thanks! So by "used to", you mean you don't think it as an additional structure on a measure space now?2012-03-29
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    @Tim: I certainly do. Don't judge my English so strictly :) I use to use the phrase "I used to"2012-03-29
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    @Ilya: I am not to judge that :). Do you see some references that treat independence as a structure on a measure space? I am also curious what can be qualified as a structure, and what cannot be.2012-03-29

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