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
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:
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Associativity:
\( (a \cdot b) \cdot c = a \cdot (b \cdot c) \) for all \( a, b, c \in G. \) -
Identity:
There exists an element \( e \in G \) such that \( a \cdot e = a = e \cdot a \) for all \( a \in G. \) -
Inverse:
There exists \( a^{-1} \in G \) for every \( a \in G \) such that \( a \cdot a^{-1} = e = a^{-1} \cdot a. \)
Group Closure Property
Some texts include an additional fourth axiom for the closure property, stated as:
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Closure:
\( a \cdot b \in G \) for all \( a, b \in G. \)
However many texts omit this axiom because explicitly stating the closure property is redundant. The binary operation \( \cdot \) on \( G \) inherently guarantees the closure property. By definition, a binary operation on a set \( G \) is a function \[ \cdot : G \times G \to G; \; (x, y) \mapsto x \cdot y, \] which automatically implies the closure property. In practice, we should always verify that the binary operation defined does indeed have the codomain \( G \) to confirm that the closure property holds.
Abelian Group
If a group satisfies the following additional axiom, then it is called an abelian group:
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Commutativity:
\( a \cdot b = b \cdot a \) for all \( a, b \in G. \)
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:
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Associativity of addition:
\( a + (b + c) = (a + b) + c \) for all \( a, b, c \in F. \) -
Commutativity of addition:
\( a + b = b + a \) for all \( a, b \in F. \) -
Additive identity:
There is an element \( 0 \in F \) such that \( a + 0 = a \) for all \( a \in F. \) -
Additive inverse:
There exists an element \( -a \in F \) for every \( a \in F \) such that \( a + (-a) = 0. \) -
Associativity of multiplication:
\( a \cdot (b \cdot c) = (a \cdot b) \cdot c \) for all \( a, b, c \in F. \) -
Commutativity of multiplication:
\( a \cdot b = b \cdot a \) for all \( a, b \in F. \) -
Multiplicative identity:
There is an element \( 1 \in F \) such that \( a \cdot 1 = a \) for all \( a \in F. \) -
Multiplicative inverse:
There exists an element \( a^{-1} \in F \) for every \( a \neq 0 \in F \) such that \( a \cdot a^{-1} = 1. \) -
Distributivity of multiplication over addition:
\( a \cdot (b + c) = (a \cdot b) + (a \cdot c) \) for all \( a, b, c \in F. \)
Ring Closure Properties
Ring Examples
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:
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Associativity of addition:
\( a + (b + c) = (a + b) + c \) for all \( a, b, c \in F. \) -
Commutativity of addition:
\( a + b = b + a \) for all \( a, b \in F. \) -
Additive identity:
There is an element \( 0 \in F \) such that \( a + 0 = a \) for all \( a \in F. \) -
Additive inverse:
There exists an element \( -a \in F \) for every \( a \in F \) such that \( a + (-a) = 0. \) -
Associativity of multiplication:
\( a \cdot (b \cdot c) = (a \cdot b) \cdot c \) for all \( a, b, c \in F. \) -
Commutativity of multiplication:
\( a \cdot b = b \cdot a \) for all \( a, b \in F. \) -
Multiplicative identity:
There is an element \( 1 \in F \) such that \( a \cdot 1 = a \) for all \( a \in F. \) -
Multiplicative inverse:
There exists an element \( a^{-1} \in F \) for every \( a \neq 0 \in F \) such that \( a \cdot a^{-1} = 1. \) -
Distributivity of multiplication over addition:
\( a \cdot (b + c) = (a \cdot b) + (a \cdot c) \) for all \( a, b, c \in F. \)
Field Closure Properties
Some texts include the following additional axioms for the closure properties:
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Closure under additon:
\( a + b \in F \) for all \( a, b \in F. \) -
Closure under multiplication:
\( a \cdot b \in F \) for all \( a, b \in F. \)
However many texts omit these two axioms because these axioms are already implied by the fact that \( + \) and \( \cdot \) are binary operations on \( F. \)
Alternative Definition of Field
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.