Algorithmic theory of integers

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Integer numbers form a set, usually denoted integer (or $Z$). The set contains a non-empty subset $N$ (of non-negative integer numbers aka natural numbers) and equipped with two binary operations , of addition and multiplication, usually denoted, $+$ i $\cdot$. These operations fulfill the following axioms:

• (Commutativity) For every pair of integer numbers $a, b$ the following equalities hold

$a+b=b+a\qquad$ and $\qquad a\cdot b = b \cdot a$,

• (Associativity) For every triplet of integer numbers $a,b, c$ the following equalities hold

$(a+(b+c))=((a+b)+c) \qquad$ and $\qquad (a\cdot(b\cdot c))=((a\cdot b)\cdot c)$,

• (Distributivity) For every triplet of integer numbers the following equality holds

$(a+b)\cdot c =a\cdot c + b\cdot c$

• (Units) There exist integer numbers 0 i 1 such that, for every $a$ the following equalities hold

$a+0=a\qquad$ and $\qquad a\cdot 1= a$,

• (Closure in $N$) If $a$ and $b$ are non-negative integer numbers then numbers $a+b$ and $a\cdot b$ are also non-negative integer numbers,
• (Additive inverse) For every integer number $a$, exists integer number $-a$, such that the following equality holds

$a+-a =0$

• (Trichotomy) For every integer number $a$ exactly one of three relations holds: either a) number $a$ is non-negative, or b) number $a$ is zero $a=0$ or c) number $-a$ jis non-negative,
• (Non-negative integers) The set of non-negative integers satisfies the axioms of natural numbers, it is the same set $N$. See Algorithmic_theory_of_natural _numbers.

Note, axiom S of natural numbers is an algorithmic formula.

• (Ordering relation) a < b if and only if the number $b+-a$ belongs to the set $N$.