Algebraic Structures

By Susam Pal on 21 Oct 2024 [draft]

An algebraic structure consists of a nonempty set, a collection of operations on the set, and a finite set of axioms that the operations must satisfy. This page describes some commonly used algebraic structures.

Contents

Magma

Magma Definition

A magma is an ordered pair \( (M, \cdot) \) where \( M \) is a set and \( \cdot \) is a binary operation on \( M. \)

Magma Closure Property

Some texts explicitly specify the closure property, i.e., for all \( a, b \in M, \) we must have \( a \cdot b \in M. \) However, stating this propery explicity is redundant because a binary operation is closed by definition. A binary operation \( \cdot \) on a set \( M \) is defined to be a function \[ \cdot : M \times M \to M; \; (x, y) \mapsto x \cdot y. \] This definition automatically implies the closure property. In practice, we should always verify that the binary operation defined does indeed have the codomain \( M \) to confirm that the closure property holds.

Magma Terminology

A magma is also known as binar. The term groupoid was used to refer to magma in early texts but this term is also used in category theory to refer to a category in which all morphisms are invertible. This term has also been used in early texts to refer to a set with a partial binary operation. Due to the ambiguity associated with this term, it is better to use the term magma to refer to a set along with its binary operation.

Magma Variations

A magma may satisfy the following additional properties:

A magma satisfying the commutativity property is known as a commutative magma. A magma satisfying the identity property is known as a unital magma. A magma satisfying both properties is known as commutative unital magma.

Semigroup

Semigroup Definition

A semigroup is an ordered pair \( (S, \cdot) \) where \( S \) is a set and \( \cdot \) is a binary operation on \( M \) satisfying the following property:

Associativity: For all \( a, b, c \in S, \) we have \( (a \cdot b) \cdot c = a \cdot (b \cdot c). \)

Semigroup Closure Property

Some texts explicitly specify the closure property, i.e., for all \( a, b \in S, \) we must have \( a \cdot b \in S. \) However, stating this propery explicity is redundant because a binary operation is closed by definition.

Semigroup Variations

A semigroup may satisfy the following additional properties:

A semigroup satisfying the commutativity property is known as a commutative semigroup or abelian semigroup. A semigroup satisfying the identity property is known as a monoid. A semigroup satisfying both properties is known as a commutative monoid. Monoid is discussed in detail in the next section.

Monoid

Monoid Definition

A monoid is an ordered pair \( (S, \cdot) \) where \( S \) is a set and \( \cdot \) is a binary operation on \( S \) satisfying the following properties:

  1. Associativity: For all \( a, b, c \in S, \) we have \( (a \cdot b) \cdot c = a \cdot (b \cdot c). \)

  2. Identity: There exists an element \( e \in S \) such that for all \( a \in S, \) we have \( a \cdot e = a = e \cdot a. \)

Monoid Closure Property

Some texts explicitly specify the closure property, i.e., for all \( a, b \in S, \) we must have \( a \cdot b \in S. \) However, stating this propery explicity is redundant because a binary operation is closed by definition.

Monoid Variations

A monoid may satisfy the following additional property:

Commutativity: For all \( a, b \in S, \) we have \( a \cdot b = b \cdot a. \)

A monoid satisfying the commutative property is known as commutative monoid or, less commonly, abelian monoid.

Group

Group Axioms

A group is an ordered pair \( (G, \cdot) \) where \( G \) is a set and \( \cdot \) is a binary operation on \( G \) satisfying the following properties:

  1. Associativity: For all \( a, b, c \in G, \) we have \( (a \cdot b) \cdot c = a \cdot (b \cdot c). \)

  2. Identity: There exists an element \( e \in G \) such that for all \( a \in G, \) we have \( a \cdot e = a = e \cdot a. \)

  3. Inverse: For each \( a \in G, \) there exists \( a^{-1} \in G \) such that \( a \cdot a^{-1} = e = a^{-1} \cdot a. \)

Group Closure Property

Some texts explicitly specify the closure property, i.e., for all \( a, b \in G, \) we must have \( a \cdot b \in G. \) However, stating this propery explicity is redundant because a binary operation is closed by definition.

Group Variations

If a group satisfies the following additional axiom, then it is called an abelian group:

Group Terminology

The binary operation \( \cdot \) is called the group operation or group law. The set \( G \) is called the underlying set of the group. The set \( G \) is also called a group under the binary operation.

Group Examples

The ordered pair \( (\mathbb{Z}, +) \) forms a group, i.e., the set of integers form a group under addition. In fact, it is an abelian group. However it does not form a group under multiplication, i.e., \( (\mathbb{Z}, \cdot) \) does not form a group since there is no multiplicative inverse of \( 2 \) in the set of integers.

Ring

Ring Axioms

A ring is a triple \( (R, +, \cdot), \) where \( R \) is a set, and \( + \) and \( \cdot \) are binary operations on \( R, \) called addition and multiplication respectively, that satisfies the following properties:

  1. Associativity of addition: For all \( a, b, c \in R, \) we have \( a + (b + c) = (a + b) + c. \)

  2. Commutativity of addition: For all \( a, b \in R, \) we have \( a + b = b + a. \)

  3. Additive identity: There exists an element \( 0 \in R \) such that for all \( a \in R, \) we have \( a + 0 = a. \)

  4. Additive inverse: For each \( a \in R, \) there exists an element \( -a \in R \) such that \( a + (-a) = 0. \)

  5. Associativity of multiplication: For all \( a, b, c \in R, \) we have \( a \cdot (b \cdot c) = (a \cdot b) \cdot c. \)

  6. Left distributivity of multiplication over addition: For all \( a, b, c \in R, \) we have \( a \cdot (b + c) = (a \cdot b) + (a \cdot c). \)

  7. Right distributivity of multiplication over addition: For all \( a, b, c \in R, \) we have \( (b + c) \cdot a = (b \cdot a) + (c \cdot a). \)

The ring defined in this section is also called noncommutative, non-unital ring or noncommutative pseudo-ring or rng (i.e., "ring" without "i", signifying the missing requirement for multiplicative identity).

Ring Closure Properties

Some texts include the following additional axioms for the closure properties:

  1. Closure under additon: For all \( a, b \in R, \) we have \( a + b \in R. \)

  2. Closure under multiplication: For all \( a, b \in R, \) we have \( a \cdot b \in R. \)

However many texts omit these two axioms because these axioms are already implied by the fact that \( + \) and \( \cdot \) are binary operations on \( R. \)

Ring Variations

In addition to the ring axioms defined earlier, a ring may be defined to have one or more of the following optional properties:

  1. Multiplicative identity: There exists an element \( 1 \in R \) such that for all \( a \in R, \) we have \( a \cdot 1 = a = 1 \cdot a. \)

  2. Commutativity of multiplication: For all \( a, b \in R, \) we have \( a \cdot b = b \cdot a. \)

A ring satisfying the first property mentioned here is called unit ring or unital ring or ring with unity or ring with identity or ring with 1. A ring satisfying the second property mentioned here is called commutative ring.

Many texts define ring to have the multiplicative identity property, so such texts mean unit ring when they refer to ring. Some texts define ring to have both the multiplicative identity property and the commutative property, so such texts mean commutative unit ring when they refer to ring.

Note that when a ring is defined to be commutative, the right distributivity property does not need to be explicitly spelled out since that is automatically implied by commutativity of multiplication.

Ring Examples

The triple \( (\mathbb{Z}, +, \cdot) \) is an obvious example of a ring. In fact, this is a commutative unit ring.

The triple \( (2\mathbb{Z}, +, \cdot) \) where \( 2\mathbb{Z} \) represents the set of all even integers, is a commutative non-unital ring.

Field

Field Axioms

A field is a triple \( (F, +, \cdot), \) where \( F \) is a set, and \( + \) and \( \cdot \) are binary operations on \( F, \) called addition and multiplication respectively, that satisfies the following properties:

  1. Associativity of addition: For all \( a, b, c \in F, \) we have \( a + (b + c) = (a + b) + c. \)

  2. Commutativity of addition: For all \( a, b \in F, \) we have \( a + b = b + a. \)

  3. Additive identity: There exists an element \( 0 \in F \) such that for all \( a \in F, \) we have \( a + 0 = a. \)

  4. Additive inverse: For each \( a \in F, \) there exists an element \( -a \in F \) such that \( a + (-a) = 0. \)

  5. Associativity of multiplication: For all \( a, b, c \in F, \) we have \( a \cdot (b \cdot c) = (a \cdot b) \cdot c. \)

  6. Commutativity of multiplication: For all \( a, b \in F, \) we have \( a \cdot b = b \cdot a. \)

  7. Multiplicative identity: There exists an element \( 1 \in F \) such that for all \( a \in F, \) we have \( a \cdot 1 = a. \)

  8. Multiplicative inverse: For each \( a \neq 0 \in F, \) there exists an element \( a^{-1} \in F \) such that \( a \cdot a^{-1} = 1. \)

  9. Distributivity of multiplication over addition: For all \( a, b, c \in F, \) we have \( a \cdot (b + c) = (a \cdot b) + (a \cdot c). \)

Field Closure Properties

Some texts include the following additional axioms for the closure properties:

  1. Closure under additon: For all \( a, b \in F \) we have \( a + b \in F. \)

  2. Closure under multiplication: For all \( a, b \in F, \) we have \( a \cdot b \in F. \)

However many texts omit these two axioms because these axioms are already implied by the fact that \( + \) and \( \cdot \) are binary operations on \( F. \)

Field Definition Alternatives

A field is a triple consisting of a set and two binary operations, called addition and multiplication, such that the set is an abelian group under addition and the nonzero elements (elements that are unequal to the additive identity) of the set form an abelian group under multplication, and multiplication distributes over addition.

Field Examples

The triple \( (\mathbb{Q}, +, \cdot) \) is a field where \( \mathbb{Q} \) denotes the set of all rational numbers. However \( \mathbb{Z}, +, \cdot \) is not a field because there is no multiplicative inverse of \( 2 \) in the set of integers.

In fact, a field is informally described as a set along with two binary operations satisfying properties analogous to those of addition and multiplication of rational numbers or real numbers.

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