Structure-preserving maps: homomorphisms and isomorphisms

Se basa en

Requisitos previos obligatorios, hasta el fondo: 8

Aprenderás a

• 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.

Definición

• 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.

Teorema

• 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.

Errores frecuentes

• 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.

Conduce a

Descubrimientos opcionales

• 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

Itinerarios de aprendizaje

• Accelerated secondary progression

Lecturas complementarias

• F. William Lawvere and Stephen H. Schanuel, Conceptual Mathematics: A First Introduction to Categories (1997)