[Math] Mathematical structures and signature

meta-mathmodel-theory

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!

Best Answer

  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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