**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:

*Set-theoretical:*relations (order, equivalence, etc.)*Algebraic:*groups, algebras, fields, vector spaces etc.*Geometrical:*topology, metric, smooth structure etc.*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:**

if my impression is right?

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.