Algebraic Structures
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
Magma Closure Property
Some texts explicitly specify the closure property, i.e., for all
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:
-
Commutativity: For all
we have -
Identity: There exists an element
such that for all we have
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
Associativity: For all
Semigroup Closure Property
Some texts explicitly specify the closure property, i.e., for all
Semigroup Variations
A semigroup may satisfy the following additional properties:
-
Commutativity: For all
we have -
Identity: There exists an element
such that for all we have
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
-
Associativity: For all
we have -
Identity: There exists an element
such that for all we have
Monoid Closure Property
Some texts explicitly specify the closure property, i.e., for all
Monoid Variations
A monoid may satisfy the following additional property:
Commutativity: For all
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
-
Associativity: For all
we have -
Identity: There exists an element
such that for all we have -
Inverse: For each
there exists such that
Group Closure Property
Some texts explicitly specify the closure property, i.e., for all
Group Variations
If a group satisfies the following additional axiom, then it is called an abelian group:
-
Commutativity: For all
we have
Group Terminology
The binary operation
Group Examples
The ordered pair
Ring
Ring Axioms
A ring is a triple
-
Associativity of addition: For all
we have -
Commutativity of addition: For all
we have -
Additive identity: There exists an element
such that for all we have -
Additive inverse: For each
there exists an element such that -
Associativity of multiplication: For all
we have -
Left distributivity of multiplication over addition: For all
we have -
Right distributivity of multiplication over addition:> For all
we have
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:
-
Closure under additon: For all
we have -
Closure under multiplication: For all
we have
However many texts omit these two axioms because these axioms are
already implied by the fact that
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:
-
Multiplicative identity: There exists an element
such that for all we have -
Commutativity of multiplication: For all
we have
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
The triple
Field
Field Axioms
A field is a triple
-
Associativity of addition: For all
we have -
Commutativity of addition: For all
we have -
Additive identity: There exists an element
such that for all we have -
Additive inverse: For each
there exists an element such that -
Associativity of multiplication: For all
we have -
Commutativity of multiplication: For all
we have -
Multiplicative identity: There exists an element
such that for all we have -
Multiplicative inverse: For each
there exists an element such that -
Distributivity of multiplication over addition: For all
we have
Field Closure Properties
Some texts include the following additional axioms for the closure properties:
-
Closure under additon: For all
we have -
Closure under multiplication: For all
we have
However many texts omit these two axioms because these axioms are
already implied by the fact that
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
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.