4.1 Structure vs Instance

A structure is a pattern. An instance is a concrete realization of that pattern.

This distinction is central to mathematical reasoning. It lets us prove results once at the structural level and then apply them to many examples.

A group is a structure. The integers under addition are an instance of that structure. The nonzero rational numbers under multiplication are another instance. The symmetries of a square are another.

Structure	Instance
Group	$(\mathbb{Z}, +)$
Group	Nonzero rational numbers under multiplication
Group	Symmetries of a square
Vector space	$\mathbb{R}^n$ over $\mathbb{R}$
Graph	A road network
Topological space	The real line with its usual open sets

The structure specifies what data and laws are required. The instance supplies actual objects satisfying those requirements.

For a group, the structure requires:

Part	Requirement
Carrier	A set $G$
Operation	A binary operation $\ast : G \times G \to G$
Associativity	$(a \ast b) \ast c = a \ast (b \ast c)$
Identity	There exists $e \in G$ such that $e \ast a = a \ast e = a$
Inverses	For every $a \in G$ , there exists $a^{-1}$ with $a \ast a^{-1} = e$

An instance is valid only if it satisfies the laws.

For example, $(\mathbb{Z}, +)$ is a group. The carrier is $\mathbb{Z}$ . The operation is addition. The identity is $0$ . The inverse of $n$ is $-n$ .

By contrast, $(\mathbb{N}, +)$ is not a group if $\mathbb{N} = {0,1,2,\dots}$ , because most elements do not have additive inverses inside $\mathbb{N}$ .

This illustrates a general rule: having similar notation does not guarantee the same structure.

The expression $a + b$ may appear in both $\mathbb{Z}$ and $\mathbb{N}$ , but the structural laws differ.

Why the distinction matters

Structural reasoning works at the level of the pattern.

If a theorem is proved for all groups, it applies to every valid group instance.

Example:

In any group, the identity element is unique.

Once this theorem is proved structurally, we do not need to reprove it for integers, matrices, permutations, or symmetries. Each instance inherits the result.

This is the main benefit of abstraction.

Level	Work done
Structure	Prove general theorems
Instance	Verify the axioms
Application	Use inherited results

The cost of using a structure is that an instance must be checked. Before applying a group theorem, we must know that the object is a group. Before applying a vector-space theorem, we must know that the object satisfies the vector-space axioms.

Instances can carry many structures

The same object can instantiate several structures.

The real numbers $\mathbb{R}$ can be viewed as:

View	Structure
$(\mathbb{R}, +)$	Additive group
$(\mathbb{R}, +, \cdot)$	Field
$(\mathbb{R}, <)$	Ordered set
$(\mathbb{R}, d)$	Metric space
$\mathbb{R}$ with open intervals	Topological space
$\mathbb{R}$ over $\mathbb{Q}$	Vector space

Each structure supports different theorems.

If we treat $\mathbb{R}$ as a field, we can discuss multiplicative inverses for nonzero elements. If we treat it as a topological space, we can discuss open sets and continuity. If we treat it as an ordered set, we can discuss upper bounds and monotonicity.

The instance alone does not determine the active structure. The text must state it or make it clear from context.

Forgetting structure

Sometimes we intentionally ignore part of a structure. This is called forgetting structure.

A field can be viewed as a ring. A ring can be viewed as an additive group. A metric space can be viewed as a topological space.

Richer structure	Forgotten structure
Field	Ring
Ring	Abelian group under addition
Metric space	Topological space
Vector space	Set
Group	Set

Forgetting structure makes fewer operations available. It may also make more maps valid.

A linear map preserves vector-space structure. A function between the underlying sets need not preserve anything.

T : V \to W

as a linear map is more restrictive than

f : |V| \to |W|

as a plain function between underlying sets.

The notation $|V|$ is sometimes used to mean the underlying set of $V$ after forgetting its extra structure.

Adding structure

The opposite process is adding structure. A set can be given a topology, a group operation, an order, or a metric.

For example, the set $\mathbb{R}$ can be equipped with the usual topology, the discrete topology, or other topologies. These produce different topological spaces on the same underlying set.

Underlying set	Added structure	Result
$\mathbb{R}$	Usual topology	Standard real line
$\mathbb{R}$	Discrete topology	Discrete space
$\mathbb{Z}$	Addition	Additive group
A finite set	Edges	Graph
A set of points	Distance function	Metric space

Adding structure increases what can be said. It also increases what must be preserved by maps.

Models as instances

In logic, an instance of a theory is often called a model.

A theory gives axioms. A model gives objects that satisfy those axioms.

M \models T

means that the structure $M$ satisfies the theory $T$ .

For example, the group axioms define a theory. Every group is a model of that theory.

This connects structural thinking with formal logic. A structure is not only a pattern in ordinary mathematics. It can also be an interpretation of a formal language.

Canonical and noncanonical instances

Some instances arise naturally. Others require choices.

The vector space $k^n$ is a standard model of an $n$ -dimensional vector space over $k$ . Every $n$ -dimensional vector space over $k$ is isomorphic to $k^n$ , but the isomorphism usually depends on a choice of basis.

V \cong k^n

This is true structurally, but not canonical unless a basis has been chosen.

The distinction matters.

Statement	Meaning
$V \cong k^n$	There exists an isomorphism
$V = k^n$	The objects are literally equal
A basis identifies $V$ with $k^n$	A choice gives a concrete representation

Structural sameness does not automatically give literal identity.

Representation vs structure

An instance often has a representation. The representation is how we store or write it. The structure is what the representation means.

A matrix can represent a linear map after bases are chosen.

A \in M_{m \times n}(k)

represents a map

T : k^n \to k^m

Changing bases changes the matrix but not the underlying linear map.

Thus, a matrix representation is coordinate-dependent. The linear map is structural.

Object	Representation	Structural meaning
Vector	Coordinate column	Element of a vector space
Matrix	Array of scalars	Linear transformation
Graph	Adjacency list	Vertices with adjacency relation
Polynomial	Coefficient list	Formal expression or function
Topological space	List of open sets	Structure of neighborhoods

Confusing representation with structure leads to fragile reasoning.

Transport of structure

An isomorphism allows structure to be transported from one object to another.

If $f : A \to B$ is a bijection and $A$ has an operation $\ast$ , we can define an operation on $B$ by transporting through $f$ :

b_1 \star b_2 = f(f^{-1}(b_1) \ast f^{-1}(b_2)).

This makes $B$ structurally equivalent to $A$ .

Transport of structure explains why isomorphic objects share the same structural properties. It also explains why equality is often too strict for mathematics.

Practical rule

When entering a new topic, separate four layers.

Layer	Question
Underlying objects	What are the elements?
Structure	What operations, relations, or laws are present?
Representation	How are objects written or stored?
Equivalence	When are two instances considered the same?

This prevents category errors.

For example, when studying graphs, ask whether vertices are labeled or unlabeled. When studying vector spaces, ask whether a basis is fixed. When studying topological spaces, ask which topology is being used.

The same data can support different structures. The same structure can have different representations. The same structural object can have many isomorphic instances.

Mathematics becomes clearer when these levels remain separate.