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
- Can structures defined as a set of subsets, such as $\sigma$-algebra, topology, be described as signature-structures?
- Can structures defined as a mapping from the set to another set, such as metric, measure, norm, inner product, be described as signature-structures?
Are signature-structures special kinds of structures defined only by operations and/or relations?
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?
Fundamentally, is "mathematical structure" a concept of model theory, category theory, or some other theory?
Thanks and regards!