Axioms of abstract algebra

axiom	a	aa	aaa
closure	C				CLOS C
associativity	A				assoc A
identity	I				id I
invertability	R				inv R
commutativity	M				comm M
absorption	B				absor B
idempotence	P				P

algebra = binop (binary operation) + (carrier) set + axioms

Axioms:

• closure (totality) CLOS C

• associativity assoc A

• identity id I

• invertability inv R

• commutativity comm M

• absorption (annihilation): B

  • ∃0 ∈ T. ∀a ∈ T. a • 0 = 0

• idempotence P

  • A ∪ A = A

  • p ∨ p = p

Elements:

• identity element: only 1 in the entire set

• inverse element: each element has one

• element one (addition identity): only 1

• element zero (multiplicative identity): only 1

  • annihilator element, p ∧ F = F regardless of p,

  • p ∧ F = F no matter what p is

  • S ∩ ∅ = ∅ intersection of any set with ∅ is ∅

  • x * 0 = 0 multiplying by 0 annihilates all other integers giving zero

Two binary operations, ⩘ and ⩗, are said to be connected by the absorption law if: a ⩘ (a ⩗ b) = a ⩗ (a ⩘ b) = a

The axiom of absorption states that there exists an element in the carrier set, called element zero, such that when any other element is combined with it, the result is always zero. Unlike the identity/neutral element, commonly called one, which leaves the other element intact, "zero" completely annihilates everything it comes in contact with.

∃0 ∈ T. ∀x ∈ T. x • 0 = 0 ∃1 ∈ T. ∀x ∈ T. x • 1 = x = 1 • x