User talk:MFH/math/counter-examples

On this page, I collect counter-examples, both from algebra and analysis, and of varying degree of sophistication.

Algebra

Multiplicative inverse but zero divisor

The sedenions have a multiplicative identity element $e_{0}$ and multiplicative inverses but they are not a division algebra because they have zero divisors. This means that two non-zero sedenions can be multiplied to obtain zero: an example is ($e_{3}$ + $e_{10}$)($e_{6}$ − $e_{15}$). All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction contain zero divisors.

Morphisms

From Morphism#Some_specific_morphisms:

Def.: Monomorphisms f: X → Y are such that for all g, h: Z → X, f ∘ g = f ∘ h ⇒ g = h. (regular (or simplifiable) to the left)

Prop.: Morphisms which have a left inverse (or retraction) g: Y → X such that g ∘ f = id_X are mono, but not the converse.

Counter-Example: A mono which has no left inverse: f: x ↦ x + 1 in ℕ? No, you can take g: x ↦ min(x - 1, 0). Indeed, g ∘ f = id.

Or f: x ↦ 2x in ℕ? No, you can take g: x ↦ floor(x/2), to get again g ∘ f = id.

Def: A mono which has a left inverse is called a split mono.

Prop.: Functions which have a left inverse are injective.

The condition of being an injection is stronger than that of being a monomorphism, but weaker than that of being a split monomorphism.

Counter-Examples : (1) mono which is not injection ; (2) injection that is not a split.

(2) this injection must be a function with left inverse which is not a mono. But a mono is defined as ..., so f must not be a morphism....?

Analysis

(to be written)