Algebraic Structures of Sets

Consider a general mapping f : A → B. Then we use the wording • domain of f := A

• range of f := B

• image of f := {f(x) | x ∈ A} Furthermore, the mapping f is called

• injective :⇔ ∀ x, y ∈ A : f(x) = f(y) ⇒ x = y

• surjective :⇔ ∀ y ∈ B ∃ x ∈ A : f(x) = y

• bijective :⇔ f is surjective and injective

Now, we will consider mappings f : G × G → G. In place of f(x, y) we will write x ∘ y. Generally, ×, ∘, ·, + and ∗ denote binary operators.

Let (F, +) and (G, ∘) or even (F, +, ·) and (G, ∘, ×) be the same algebraic structure, e.g. a group, a ring, a field, e.t.c. . A mapping f : F → G is called

• homomorphism :⇔ ∀ x, y ∈ F : f(x + y) = f(x) ∘ f(y) and f(x · y) = f(x) × f(y)

• endomorphism :⇔ f is a homomorphism and F = G

• isomorphism :⇔ f is a homomorphism and f is bijective

• automorphism :⇔ f is a isomorphism and F = G

There are several importants rules and notions for a binary operator:

• associative ∘ is called associative :⇔ ∀ x, y, z ∈ G : (x ∘ y) ∘ z = x ∘ (y ∘ z)

• commutative ∘ is called commutative :⇔ ∀ x, y ∈ G : x ∘ y = y ∘ x

• zero element (annihilator)

0 ∈ G is called zero element regarding ∘ :⇔ ∀ x ∈ G : 0 ∘ x = x ∘ 0 = 0 • identity

e ∈ G is an identity regarding ∘ :⇔ ∀ x ∈ G : e ∘ x = x ∘ e = x

• inverse element of x y ∈ G is called the inverse of x ∈ G :⇔ ∃ y ∈ G : (x∘y)∘x = x∘(y∘x) = x. If there is an identity e ∈ G, we have x ∘ y = y ∘ x = e.

Definitions of algebraic structures

• groupoid (G, ∘) is called groupoid :⇔ ∀ x, y ∈ G : x ∘ y ∈ G

• semi-group (G, ∘) is called semi-group :⇔ (G, ∘) is a groupoid and ∘ is associative.

• monoid (G, ∘) is called monoid :⇔ (G, ∘) is a semi-group and ∃ an identity in G regarding ∘.

• quasigroup (G, ∘) is called quasigroup :⇔ (G, ∘) is a groupoid and ∀ a, b ∈ G ∃! x, y ∈ G : a ∘ x = b y ∘ a = b.

• loop (G, ∘) is called loop :⇔ (G, ∘) is a quasigroup and ∃ an identity in G regarding ∘.

• group (G, ∘) is called group :⇔ (G, ∘) is a loop and ∘ is associative.

• abelian group (G, ∘) is called abelain group :⇔ (G, ∘) is a group and ∘ is commutative.

• lattice (G, ∘, ×) is called a lattice :⇔

1. (G, ∘) is a commutative semi-group,

2. (G, ×) is a commutative semi-group

3. ∀ x, y ∈ G : x × (x ∘ y) = x, x ∘ (x × y) = x

• complemented lattice (G, ∘, ×) is called a complemented lattice :⇔

1. (G, ∘, ×) is a lattice and

2. ∃ 0 ∈ G ∀ x ∈ G : 0 ∘ x = 0, 0 × x = x

3. ∃ 1 ∈ G ∀ x ∈ G : 1 ∘ x = x, 1 × x = 1

• distributive lattice (G, ∘, ×) is called a distributive lattice :⇔ (G, ∘, ×) is a lattice and ∀ x, y, z ∈ G : x × (y ∘ z) = (x × y) ∘ (x × z), (z ∘ y) × x = (z × x) ∘ (y × x)

• boolean lattice (boolean algebra) (G, ∘, ×) is called a boolean lattice :⇔ (G, ∘, ×) is a complemented lattice and a distributive lattice

• semi-ring (G, ∘, ×) is called a semi-ring :⇔

1. (G, ∘) is a semi-group,

2. (G, ×) is a semi-group

3. ∀ x, y, z ∈ G : x×(y∘z) = (x×y)∘(x×z), (z∘y)×x = (z×x)∘(y×x)

• near-ring (G, ∘, ×) is called a near-ring :⇔

1. (G, ∘) is a group,

2. (G, ×) is a semigroup

3. ∀ x, y, z ∈ G : (z ∘ y) × x = (z × x) ∘ (y × x)

• non-associative ring (G, ∘, ×) is called a non-associative ring :⇔

1. (G, ∘) is an abelian group

2. (G, ×) is a groupoid

3. ∀ x, y, z ∈ G : x×(y∘z) = (x×y)∘(x×z), (z∘y)×x = (z×x)∘(y×x)

• ring (G, ∘, ×) is called ring :⇔ (G, ∘, ×) is a non-associative ring and × is associative.

• commutative ring (G, ∘, ×) is called commutative ring :⇔ (G, ∘, ×) is a ring and × is commutative.

• ring with identity (G, ∘, ×) is called ring with identity :⇔ (G, ∘, ×) is a ring and there is an identity in G regarding ×.

• commutative ring with identity (G, ∘, ×) is called comm. ring with identity :⇔ (G, ∘, ×) is a ring with identity and × is commutative.

• division ring (G, ∘, ×) is called division ring :⇔ (G, ∘, ×) is a ring and ∀ x, y ∈ G : x × y = 0 ⇔ x = 0 or y = 0 (0 must be the identity regarding ∘).

• division-algebra (G, ∘, ×) is called division-algebra :⇔ (G, ∘, ×) is a ring with identity and ∀ x ∈ G\{0} ∃ y ∈ G : x × y = y × x = 1 . Here 0 denotes the identitiy regarding ∘ in G and 1 the identity regarding ×.

• field (G, ∘, ×) is called field :⇔ (G, ∘, ×) is a division-algebra and × is commutative.

• modul

((F, ∘, ×), (G, +), ·) is called a modul over F :⇔ 1. (G, +) is a group (G is called the set of vectors)

2. (F, ∘, ×) is a ring with identity (F is called the set of scalars)

3. ∀ a ∈ F ∀ x ∈ G : a · x ∈ G

4. ∀ a, b ∈ F ∀ x ∈ G : (a × b) · x = a · (b · x)

5. ∀ a, b ∈ F ∀ x ∈ G : (a ∘ b) · x = (a · x) + (b · x)

6. ∀ x ∈ G : 1 · x = x with 1 the identity of F regarding ×

• vectorspace

((F, ∘, ×), (G, +), ·) is called a vectorspace over F :⇔ 0. ((F, ∘, ×), (G, +), ·) is a modul over F

1. (G, +) is a group (G is called the set of vectors)

2. (F, ∘, ×) is a field (F is called the set of scalars)

• algebra

((F, ∘, ×), (G, +, ∗), ·) is called an algebra over F :⇔ 0. ((F, ∘, ×), (G, +), ·) is a vectorspace over F

1. (G, +, ∗) is a ring (G is called the set of vectors)

2. (F, ∘, ×) is a field (F is called the set of scalars)

3. ∀ a ∈ F ∀ x, y ∈ G : a · (x ∗ y) = (a · x) ∗ y = x ∗ (a · y)

• commutative algebra

((F, ∘, ×), (G, +, ∗), ·) is called a commutative algebra over F :⇔ 0. ((F, ∘, ×), (G, +, ∗), ·) is an algebra over F

1. (G, +, ∗) is a commutative ring (G is called the set of vectors)

Bielefeld, 1998–11–10