Quasi omomorfismo

Un quasi omomorfismo è un'applicazione da $\mathbb {Z}$ in sé che può essere considerata una generalizzazione degli omomorfismi.

Definizione

Una funzione $f:\mathbb {Z} \longrightarrow \mathbb {Z}$ si dice quasi omomorfismo se $\exists k\in \mathbb {N}$ tale che $\forall n,m\in \mathbb {Z}$ :

|f(n+m)-f(n)-f(m)|\leq k.

Naturalmente se k=0 si è in presenza di un omomorfismo.

Proprietà

Sia $f:\mathbb {Z} \longrightarrow \mathbb {Z}$ un quasi omomorfismo con costante $k\in \mathbb {N}$ :

$\forall n,m\in \mathbb {Z} :|f(m\cdot n)-m\cdot f(n)|\leq (|m|+1)\cdot k$ ;
$\forall n,m\in \mathbb {Z} :|n\cdot f(m)-m\cdot f(n)|\leq (|n|+|m|+2)\cdot k$ ;
Ad ogni quasi omomorfismo corrisponde una successione di Cauchy e viceversa.^[1]

Classi di equivalenza

Si può definire una relazione tra quasi omomorfismi nel modo seguente:

Siano

f,g

quasi omomorfismi,

f\sim g\leftrightarrow \forall n\in \mathbb {Z} ,\exists k\in \mathbb {N}

tale che

|f(n)-g(n)|\leq k.

Si dimostra facilmente che $\sim$ è una relazione d'equivalenza. Si dimostra anche che ad ogni classe di equivalenza di quasi omomorfismi $[f]_{\sim }$ corrisponde una classe di successioni di Cauchy $[a_{n}]$ . Con questo risultato si scopre che è possibile costruire l'insieme dei numeri reali $\mathbb {R}$ a partire da $\mathbb {Z}$ utilizzando classi di quasi omomorfismi.^[1]

Note

^ ^a ^b James Douglas, Rony Kirollos, Ben Odgers, Ross Street e Nguyen Hanh Vo, The Efficient Real Numbers (PDF), su web.science.mq.edu.au, Università Macquarie, febbraio 2004.

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