From Wikipedia:

In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach (or relate) to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance.

A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, equivalence relations, differential structures, and categories.

From Planetmath:

Let $\tau$ be a signature. A $\tau$-structure $\mathcal{A}$ comprises of a set $A$, called the universe (or underlying set or domain) of $\mathcal{A}$, and an interpretation of the symbols of $\tau$ as follows:

  • for each constant symbol $c\in\tau$, an element $c^A\in A$;
  • for each $n$-ary function symbol $f\in\tau$, a function (or operation) $f^A:A^n\rightarrow A$;
  • for each $n$-ary relation symbol $R\in\tau$, a $n$-ary relation $R^A$ on $A$.

I was wondering

  1. Can structures defined as a set of subsets, such as $\sigma$-algebra, topology, be described as signature-structures?
  2. Can structures defined as a mapping from the set to another set, such as metric, measure, norm, inner product, be described as signature-structures?
  3. Are signature-structures special kinds of structures defined only by operations and/or relations?

  4. In the Wikipedia quote "a structure on a set, or more generally a type", does it mean a structure is also called a type, or the underlying set is a type?

  5. Fundamentally, is "mathematical structure" a concept of model theory, category theory, or some other theory?

Thanks and regards!

  • $\begingroup$ By interpreting everything in a formal set theory, the absurdly open-ended definition of Wikipedia can be accommodated. $\endgroup$ – André Nicolas Mar 28 '12 at 23:40
  • $\begingroup$ Thanks, @AndréNicolas! Why is Wiki absurdly open ended? Don't you think sigma algebras, topologies, metric, inner product, norm are structures? $\endgroup$ – Tim Mar 28 '12 at 23:42
  • $\begingroup$ As I read the "definition," I cannot think of anything that would not qualify. As to your specific questions, one can force (for example) a general vector space to be a first-order structure by having a two predicate symbols $V$, $S$ ($x$ is a vector, $x$ is a scalar). One can operate similarly with other notions such as inner products, even measures. So far this has not proved profitable. $\endgroup$ – André Nicolas Mar 28 '12 at 23:55
  • $\begingroup$ Thanks, @AndréNicolas! What does the order of structure mean, like first-order structure, second-order structure, zero-order structure? $\endgroup$ – Tim Mar 29 '12 at 0:17
  • 1
    $\begingroup$ First order language: variables range over objects. Second-order language: There are variables of two types, the first ranging over objects, the second ranging over sets of objects. So for example Peano's axiomatization of arithmetic was second-order, every non-empty subset has a smallest element. The first-order version has infinitely many instances of the induction axiom, one for every formula $F(x)$. The second-order version has up to isomorphism one model, the first-order version has many models. $\endgroup$ – André Nicolas Mar 29 '12 at 0:28
  1. Sure. For a good explanation, see the one here by Brian M. Scott.

  2. If the sets in question are $X$ and $Y$, these mappings are just subsets of $P(X \times Y)$. In this way, one can view these spaces as model-theoretic structures as in Brian's explanation for $(1)$. More generally, check out the introduction in the paper model theory for metric structures to see how these sorts of structures are studied.

  3. In Model Theory, structures arise from theories in a given signature. I am not sure if there is a notion of a structure existing independently from a signature.

  4. A type is something of a generalization of a structure. I really don't know anything about types, so I'm hoping somebody here can fill in this part of my answer.

  5. The term "mathematical structure" has a specific meaning in model theory. To my knowledge, that is not the case in category theory. Generally speaking, it is a philosophical idea that would be hard to pin down to any specific branch of math.


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