Grades 9–12Category TheoryPlanned

Structure-preserving maps: homomorphisms and isomorphisms

Concrete homomorphisms (maps that respect an operation, such as reduction modulo n, exp from (ℝ, +) to (ℝ>0, ×), linear maps and graph homomorphisms) and isomorphisms as structure-preserving bijections whose inverses also preserve structure, making 'the same structure' precise.

math.cat.structure-preserving-maps

Show everything needed to reach this topic

The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.

Builds on

Composition

Required prerequisites, all the way down: 8

You will learn to

Check whether a given map preserves an operation, for example reduction modulo n from (ℤ, +) to clock arithmetic.
Show that composites of structure-preserving maps preserve structure, and that the inverse of a bijective homomorphism of groups is a homomorphism.
Decide whether two small structures (clock arithmetics, graphs, symmetry patterns) are isomorphic, and explain what an isomorphism transfers between them.

Definition

A map f from (A, ∗) to (B, ⋆) preserves the operation if f(x ∗ y) = f(x) ⋆ f(y) for all x and y in A.
An isomorphism is a structure-preserving bijection whose inverse is also structure-preserving.
Two structures are isomorphic if there is an isomorphism between them.

Theorem

exp : (ℝ, +) → (ℝ>0, ×) is an isomorphism, with inverse ln.
Reduction modulo n from ℤ to the integers mod n preserves addition and multiplication but is not injective.
For groups and rings, the inverse of a bijective homomorphism is automatically a homomorphism; for ordered sets or topological spaces a structure-preserving bijection can have an inverse that is not structure-preserving.
Isomorphic structures share every property that can be expressed purely in terms of their structure.

Common misconceptions

Thinking two structures are isomorphic just because they have the same number of elements: clock arithmetic mod 4 and the symmetries of a non-square rectangle are not isomorphic.
Treating isomorphism as equality; isomorphic structures can have entirely different elements.
Assuming every structure-preserving bijection has a structure-preserving inverse.

Leads to

Optional discoveries

Groups The first isomorphism theorem identifies the image of any group homomorphism with a quotient of its domain.
Model theory Isomorphic structures satisfy the same first-order sentences, but structures satisfying the same sentences need not be isomorphic.
← Rigid motions, congruence and similarity
← Diagrams where paths agree
← Exponential and logarithmic functions
← Graphs

Learning pathways

Accelerated secondary progression