Algorithmic theory of integers

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

  • (Commutativity) For every pair of integer numbers [math]a, b[/math] the following equalities hold

[math]a+b=b+a\qquad[/math] and [math]\qquad a\cdot b = b \cdot a[/math],

  • (Associativity) For every triplet of integer numbers [math]a,b, c[/math] the following equalities hold

[math](a+(b+c))=((a+b)+c) \qquad[/math] and [math]\qquad (a\cdot(b\cdot c))=((a\cdot b)\cdot c)[/math],

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

[math](a+b)\cdot c =a\cdot c + b\cdot c[/math]

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

[math]a+0=a\qquad [/math] and [math]\qquad a\cdot 1= a[/math],

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

[math] a+-a =0 [/math]

  • (Trichotomy) For every integer number [math]a[/math] exactly one of three relations holds: either a) number [math]a[/math] is non-negative, or b) number [math]a[/math] is zero [math]a=0[/math] or c) number [math]-a[/math] jis non-negative,
  • (Non-negative integers) The set of non-negative integers satisfies the axioms of natural numbers, it is the same set [math]N[/math]. 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 [math] b+-a [/math] belongs to the set [math]N[/math].