Part 2: Cyclands and integers

(This text is a continuation of Part 1 from the same thread).

First I'll axiomatize here the cyclic groups--I'll call them cyclands (to avoid any confusion during the definition stage; later one can go back to the standard naming; the same goes for Abelian groups and minusops). Then there will come time for integers.

Cyclands

By definition, a *cycland* is an ordered triple   $(X\ -\ 1)$   such that   $X$   is an arbitrary set,   symbol   $-$   stands for a binary operation in   $X$,   $1\in X$, and   the following (algebraic induction-like) axiom holds:

$$ \left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$$

Integers

DEFINITION   Integers (i.e. the system of integers) is a cycland   $(\mathbb Z\ -\ 1)$   such that the following (additional) two axioms hold:

1. $1-x\ne x$
2. $1-x=x-1\quad\Rightarrow\quad x=1$

for arbitrary   $x\in\mathbb Z$.

Integers from scratch

To have an easy overview of the definition of integers let me list all the relevant axioms directly (without mentioning minusops and cyclands).

The system of integers is an ordered triple   $(\mathbb Z\ -\ 1)   such that the following six axioms hold:

1. $\forall_{x\ y\ z\in\mathbb Z}\quad x-(y-z)\ =\ z-(y-x)$
2. $\forall_{x\ y\in\mathbb Z}\quad x-x\ =\ y-y$
3. $\forall_{x\in\mathbb Z}\quad x-(x-x) = x$
4. $\forall_{x\in\mathbb Z}\quad 1-x\ne x$
5. $\forall_{x\in\mathbb Z}\quad \left(\left(1-x=x-1\right)\ \Rightarrow\ \left(x=1\right)\right)$
6. $\left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$

Other standard constructions or notions

We define in   $\mathbb Z$   the usual, like:

• $0\ :=\ 1-1$
• $\mathit{Neg}(x)\ :=\ 0-x$
• $x+y\ :=\ x -\mathit{Neg}(y)$

Furthermore, in every cycland, in particular in   $\mathbb Z$,   there is exactly one binary operation   $\cdot$   which has the following two properties:

• $x\cdot 1\ =\ x$
• $x\cdot(y-z)\ =\ x\cdot y - x\cdot z$

for arbitrary   $x\ y\ z\ \in\ \mathbb Z$.

Part 2: Cyclands and integers

(This text is a continuation of Part 1 from the same thread).

First I'll axiomatize here the cyclic groups--I'll call them cyclands (to avoid any confusion during the definition stage; later one can go back to the standard naming; the same goes for Abelian groups and minusops). Then there will come time for integers.

Cyclands

By definition, a *cycland* is an ordered triple   $(X\ -\ 1)$   such that   $X$   is an arbitrary set,   symbol   $-$   stands for a binary operation in   $X$,   $1\in X$, and   the following (algebraic induction-like) axiom holds:

$$ \left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$$

Integers

DEFINITION   Integers (i.e. the system of integers) is a cycland   $(\mathbb Z\ -\ 1)$   such that the following (additional) two axioms hold:

1. $1-x\ne x$
2. $1-x=x-1\quad\Rightarrow\quad x=1$

for arbitrary   $x\in\mathbb Z$.

Integers from scratch

To have an easy overview of the definition of integers let me list all the relevant axioms directly (without mentioning minusops and cyclands).

The system of integers is an ordered triple   $(\mathbb Z\ -\ 1)$   such that the following six axioms hold:

1. $ x-(y-z)\ =\ z-(y-x)$
2. $ x-x\ =\ y-y$
3. $ x-(x-x) = x$
4. $ 1-x\ne x$
5. $ \left(1-x=x-1\right)\ \Rightarrow\ \left(x=1\right)$
6. $ \left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$

for arbitrary   $x\ y\ z\ \in\ \mathbb Z$.

Other standard constructions or notions

We define in   $\mathbb Z$   the usual, like:

• $0\ :=\ 1-1$
• $\mathit{Neg}(x)\ :=\ 0-x$
• $x+y\ :=\ x -\mathit{Neg}(y)$

Furthermore, in every cycland, in particular in   $\mathbb Z$,   there is exactly one binary operation   $\cdot$   which has the following two properties:

• $x\cdot 1\ =\ x$
• $x\cdot(y-z)\ =\ x\cdot y - x\cdot z$

for arbitrary   $x\ y\ z\ \in\ \mathbb Z$.

Part 2: Cyclands and integers

(This text is a continuation of Part 1 from the same thread).

First I'll axiomatize here the cyclic groups--I'll call them cyclands (to avoid any confusion during the definition stage; later one can go back to the standard naming; the same goes for Abelian groups and minusops). Then there will come time for integers.

Cyclands

By definition, a *cycland* is an ordered triple   $(X\ -\ 1)$   such that   $X$   is an arbitrary set,   symbol   $-$   stands for a binary operation in   $X$,   $1\in X$, and   the following (algebraic induction-like) axiom holds:

$$ \left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$$

Integers

DEFINITION   Integers (i.e. the system of integers) is a cycland   $(\mathbb Z\ -\ 1)$   such that the following (additional) two axioms hold:

1. $1-x\ne x$
2. $1-x=x-1\quad\Rightarrow\quad x=1$

for arbitrary   $x\in\mathbb Z$.

Integers from scratch

To have an easy overview of the definition of integers let me list all the relevant axioms directly (without mentioning minusops and cyclands).

The system of integers is an ordered triple   $(\mathbb Z\ -\ 1)   such that the following axioms hold:

1. $\forall_{x\ y\ z\in\mathbb Z}\quad x-(y-z)\ =\ z-(y-x)$
2. $\forall_{x\ y\in\mathbb Z}\quad x-x\ =\ y-y$
3. $\forall_{x\in\mathbb Z}\quad x-(x-x) = x$
4. $\forall_{x\in\mathbb Z}\quad 1-x\ne x$
5. $\forall_{x\in\mathbb Z}\quad \left(\left(1-x=x-1\right)\ \Rightarrow\ \left(x=1\right)\right)$
6. $\left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$

(to be continued in a moment)

Part 2: Cyclands and integers

(This text is a continuation of Part 1 from the same thread).

First I'll axiomatize here the cyclic groups--I'll call them cyclands (to avoid any confusion during the definition stage; later one can go back to the standard naming; the same goes for Abelian groups and minusops). Then there will come time for integers.

Cyclands

By definition, a *cycland* is an ordered triple   $(X\ -\ 1)$   such that   $X$   is an arbitrary set,   symbol   $-$   stands for a binary operation in   $X$,   $1\in X$, and   the following (algebraic induction-like) axiom holds:

$$ \left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$$

Integers

DEFINITION   Integers (i.e. the system of integers) is a cycland   $(\mathbb Z\ -\ 1)$   such that the following (additional) two axioms hold:

1. $1-x\ne x$
2. $1-x=x-1\quad\Rightarrow\quad x=1$

for arbitrary   $x\in\mathbb Z$.

Integers from scratch

To have an easy overview of the definition of integers let me list all the relevant axioms directly (without mentioning minusops and cyclands).

The system of integers is an ordered triple   $(\mathbb Z\ -\ 1)   such that the following six axioms hold:

1. $\forall_{x\ y\ z\in\mathbb Z}\quad x-(y-z)\ =\ z-(y-x)$
2. $\forall_{x\ y\in\mathbb Z}\quad x-x\ =\ y-y$
3. $\forall_{x\in\mathbb Z}\quad x-(x-x) = x$
4. $\forall_{x\in\mathbb Z}\quad 1-x\ne x$
5. $\forall_{x\in\mathbb Z}\quad \left(\left(1-x=x-1\right)\ \Rightarrow\ \left(x=1\right)\right)$
6. $\left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$

Other standard constructions or notions

We define in   $\mathbb Z$   the usual, like:

• $0\ :=\ 1-1$
• $\mathit{Neg}(x)\ :=\ 0-x$
• $x+y\ :=\ x -\mathit{Neg}(y)$

Furthermore, in every cycland, in particular in   $\mathbb Z$,   there is exactly one binary operation   $\cdot$   which has the following two properties:

• $x\cdot 1\ =\ x$
• $x\cdot(y-z)\ =\ x\cdot y - x\cdot z$

for arbitrary   $x\ y\ z\ \in\ \mathbb Z$.

Part 2: Cyclands and integers

(This text is a continuation of Part 1 from the same thread).

First I'll axiomatize here the cyclic groups--I'll call them cyclands (to avoid any confusion during the definition stage; later one can go back to the standard naming; the same goes for Abelian groups and minusops). Then there will come time for integers.

Cyclands

By definition, a *cycland* is an ordered triple   $(X\ -\ 1)$   such that   $X$   is an arbitrary set,   symbol   $-$   stands for a binary operation in   $X$,   $1\in X$, and   the following (algebraic induction-like) axiom holds:

$$ \left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$$

Integers

DEFINITION   Integers (i.e. the system of integers) is a cycland   $(\mathbb Z\ -\ 1)   such that the following (additional) two axioms hold:

1. $1-x\ne x$
2. $1-x=x-1\quad\Rightarrow\quad x=1$

for arbitrary   $x\in\mathbb Z$.

Integers from scratch

To have an easy overview of the definition of integers let me list all the relevant axioms directly (without mentioning minusops and cyclands).

The system of integers is an ordered triple   $(\mathbb Z\ -\ 1)   such that the following axioms hold:

1. $\forall_{x\ y\ z\in\mathbb Z}\quad x-(y-z)\ =\ z-(y-x)$
2. $\forall_{x\ y\in\mathbb Z}\quad x-x\

(to be continued in a moment)

Part 2: Cyclands and integers

(This text is a continuation of Part 1 from the same thread).

First I'll axiomatize here the cyclic groups--I'll call them cyclands (to avoid any confusion during the definition stage; later one can go back to the standard naming; the same goes for Abelian groups and minusops). Then there will come time for integers.

Cyclands

By definition, a *cycland* is an ordered triple   $(X\ -\ 1)$   such that   $X$   is an arbitrary set,   symbol   $-$   stands for a binary operation in   $X$,   $1\in X$, and   the following (algebraic induction-like) axiom holds:

$$ \left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$$

Integers

DEFINITION   Integers (i.e. the system of integers) is a cycland   $(\mathbb Z\ -\ 1)$   such that the following (additional) two axioms hold:

1. $1-x\ne x$
2. $1-x=x-1\quad\Rightarrow\quad x=1$

for arbitrary   $x\in\mathbb Z$.

Integers from scratch

To have an easy overview of the definition of integers let me list all the relevant axioms directly (without mentioning minusops and cyclands).

The system of integers is an ordered triple   $(\mathbb Z\ -\ 1)   such that the following axioms hold:

1. $\forall_{x\ y\ z\in\mathbb Z}\quad x-(y-z)\ =\ z-(y-x)$
2. $\forall_{x\ y\in\mathbb Z}\quad x-x\ =\ y-y$
3. $\forall_{x\in\mathbb Z}\quad x-(x-x) = x$
4. $\forall_{x\in\mathbb Z}\quad 1-x\ne x$
5. $\forall_{x\in\mathbb Z}\quad \left(\left(1-x=x-1\right)\ \Rightarrow\ \left(x=1\right)\right)$
6. $\left(\left(1\in A\subseteq X\right)\ \ \&\ \ \left(\forall_{x\ y\in A}\ x-y\in A\right)\right)\quad\Rightarrow\quad \left(A=X\right)$

(to be continued in a moment)