What does "(additional) features" mean in the definition of a structure?

Question

I am reading the Wikipedia article on mathematical structure, and it says "a structure is a set endowed with some additional features". Features can encompass a metric, a topology, etc...

I'd like to know if there exists a rigorous definition of this meaning of structure, especially of this term "features".

Could "features" mean just a signature along with an interpretation ?

Thanks in advance.

Answer 1

Often in logic and model theory structure means Signature [see Structure (Mathematical logic)].

See e.g. Dirk van Dalen, Logic and Structure page 58:

A structure is an ordered sequence $\langle A,R_1,\ldots, R_n,F_1,\ldots,F_m,\{ c_i \mid i \in \} \rangle$, where $A$ is a non-empty set [the domain]. $R_1,\ldots,R_n$ are relations on $A$, $F_1,\ldots,F_m$ are functions on $A$, the $c_i$ are elements of $A$ [the constants].

So, we have a domain of individuals: the set $A$, and relations and operations on them.

Note that in this sense a signature is related only to the language.

The Bourbaki-like term "Mathematical structure" is usually used not in conenction with a formal language, but considering more specifically the relavnt axioms: this is IMO the sense of Wiki's statement "a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology)."

There is no contradiction, because operations and metrics are functions; also a topology is a function from the space $X$ (the domain of the structure) to the collection of subsets of $X$.