What is the meaning of "structure on a set"?

Question

Mathematical Structure:

Definition 1: "A mathematical structure is nothing but a (more or less) complicated organization of smaller, more fundamental mathematical substructures. Numbers are one kind of structure, and they can be used to build bigger structures like vectors and matrices."

Definition 2: In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.

Sets: "A Set is a well-defined collection of objects."

Question:

When the author says "structure on set" or "attach structures to sets", what does it mean? How can I "attach" structures to sets?

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

My thoughts on what "attach" is:

If I build a set, and the elements of the Set satisfies required axioms of Algebraic Structure for Ring Set then I can say that I have successfully attached a structure (Algebraic Structure (Ring Set here)) to the built set?

Answer 1

There is no real meaning to the term "attach", apart from maybe writing them as an ordered pair: $(\text{Set}, \text{Structure})$. In fact, when we say things like "$X$ is a topological space", it is a relaxation in terminology of what the topological space is: a pair $(X,\tau)$, where $X$ is nothing but a set of points, and $\tau$ is the topology.

The same applies to other structures defined on sets. "A group $X$" is short for $(X,{}\cdot{})$ where ${}\cdot{}\colon X\times X\to X$ is the product, a ring is a triple $(R,+,{}\cdot{})$, a vector space $V$ is also a triple $(V(F), +,{}\cdot{})$, etc.

A set on its own never has inherent structure, but it is cumbersome to write, say, "an inner product space $(X, \langle{}\cdot{},{}\cdot{}\rangle$)" each time, so we write "an inner product space $X$".

Answer 2

The usual method of attaching a structure to a set $A$ is defining a finite sequence whose first element is the set $A$, and the following elements are additional sets, which under a proper interpretation, can be understand as a structures on $A$.

For example, to give set $A$ the structure of a partially ordereded set you construct a pair $(A, B)$, where $B$ is a partial order on set $A$, that is $B$ is a subset of $A\times A$ is such that: $$ \forall x\in A: (x,x) \in B$$ $$ \forall x,y\in A: \big((x,y) \in B \,\land (y,x)\in B\big)\Rightarrow(x=y)$$ $$ \forall x,y,z\in A: \big((x,y) \in B \,\land (y,z)\in B\big)\Rightarrow (x,z)\in B$$ (those are the reflexivity, antisymmetry and transitivity conditions). For simplicity we can introduce a notation like $$ (x\le y ) \equiv^\text{def} \big((x,y) \in B\big)$$

All kinds of structures can be defined in this way as well.