Darasa la 9–12Category TheoryImepangwa
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
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 8 (hadi chini kabisa)
Utajifunza
- 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.
Fasili
- 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.
Nadharia
- 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.
Mawazo potofu ya kawaida
- 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.
Inaelekea kwa
Ugunduzi wa hiari
- 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
Njia za kujifunza
Usomaji zaidi
- F. William Lawvere and Stephen H. Schanuel, Conceptual Mathematics: A First Introduction to Categories (1997)