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Abstract algebra chest sheet, Cheat Sheet of Algebra

Babes-Bolyai University of Cluj-NapocaAlgebra

Abstract algebră formulas computațională

Typology: Cheat Sheet

2024/2025

Uploaded on 04/06/2025

marius-calin-2
marius-calin-2 🇷🇴

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Algebra hS, i
Carrier Set S
Operator Set ={f1,...,f
n}
Operator fi:Sm!S
Arity m2N0
Commutative Group
Magma
Semigroup
Monoid
Group
inverse
xx=xx=1
Semiring
Ring
Integral Domain
Commutative Ring
Commutative Semigroup
Commutative Monoid
Commutativity
xy=yx
Identity
1x=x1=x
Associativity
(xy)z=x(yz)
hS, {,}i
:S⇥S!S
:S⇥S!S
1.Commutative Monoid hS, i
Additive Identity 02S
2.Monoid hS, i
Multiplicative Identity 12S
3.Distributivity
x(yz)=(xy)(xz)
(yz)x=(yx)(zx)
Commutative Group
hS, i
Commutative Monoid
hS, i
No Zero Divisors
8x, y 2S\{0}:xy 6=0
Commutative Group
hS, i
hS, {}i
:S⇥S!S

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Algebra hS, i

Carrier Set S

Operator Set = {f 1 ,... , fn}

Operator fi : Sm^! S

Arity m 2 N 0

Commutative Group

Magma Semigroup Monoid Group

inverse x x = x x = 1

Semiring Ring Commutative Ring Integral Domain Field

Commutative Semigroup Commutative Monoid

Commutativity x y = y x Identity 1 x = x 1 = x Associativity (x y) z = x (y z) hS, {, }i : S ⇥ S! S : S ⇥ S! S

  1. Commutative Monoid hS, i Additive Identity 0 2 S
  2. Monoid hS, i Multiplicative Identity 1 2 S
  3. Distributivity x (y z) = (x y) (x z) (y z) x = (y x) (z x) Commutative Group hS, i Commutative Monoid hS, i No Zero Divisors 8 x, y 2 S \ { 0 } : xy 6 = 0 Commutative Group hS, i

hS, {}i

: S ⇥ S! S