Algebraic Structures

non-empty set S of objects such as numbers, functions or matrices with one or more operations $*$ .

Depending on the properties of the operations and the set, the algebraic structure can be classified into different types. which can be useful for understanding the properties of the set and the operations and deriving proofs.

Operations

In mathematics, an operation is a function which takes zero or more input values (also called "operands" or "arguments") to a well-defined output value. The number of operands is the arity of the operation.

The most commonly studied operations are binary operations (i.e., operations of arity 2), such as addition and multiplication, and unary operations (i.e., operations of arity 1), such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is a constant.

Closure

An operation $*$ is closed if any operation on elements of the set $S$ results in an element that is also in the set $S$ . More formally:

\forall a, b \in S, a * b \in S

Example

Addition and multiplication are closed operations on the set of integers. However, division is not a closed operation on the set of integers. For example:

2 \div 3 = 0.6667 \notin \mathbb{Z}

Binary Operation

More specifically, a binary operation on a set is a binary operation whose two domains and the codomain are the same set, i.e the operation takes two elements from the set and returns an element from the set (closure).

N-ary Operation

Officially doesn't have to be closed.

Properties of Operations

Commutativity

An operation $*$ is commutative if for all $a, b \in S$ :

a * b = b * a

In other words, the order of the operands does not matter.

Example

Addition and multiplication are commutative operations. For example:

\begin{align*} 2 + 3 &= 3 + 2 = 5 \\ 2 \cdot 3 &= 3 \cdot 2 = 6 \end{align*}

Subtraction and division are not commutative operations. For example:

\begin{align*} 2 - 3 &\neq 3 - 2 \\ -1 &\neq 1 \\ 2 \div 3 &\neq 3 \div 2 \\ 0.6667 &\neq 1.5 \end{align*}

Associativity

An operation $*$ is associative if for all $a, b, c \in S$ :

a * (b * c) = (a * b) * c

In other words, the grouping of the operands does not matter. So you can group the operands in any way you like as long as the order of the operands is preserved and there are no other operations in between.

Example

Addition and multiplication are associative operations. For example:

\begin{align*} 2 + (3 + 4) &= (2 + 3) + 4 = 9 \\ 2 \cdot (3 \cdot 4) &= (2 \cdot 3) \cdot 4 = 24 \end{align*}

Subtraction and division are not associative operations. For example:

\begin{align*} 2 - (3 - 4) &\neq (2 - 3) - 4 \\ 3 &\neq -5 \\ 2 \div (3 \div 4) &\neq (2 \div 3) \div 4 2.666 &\neq 0.166 \end{align*}

Distributivity

An operation $*$ is distributive with respect to another operation $+$ if for all $a, b, c \in S$ :

a * (b + c) = (a * b) + (a * c)

and

(b + c) * a = (b * a) + (c * a)

In other words, the operation "distributes" itself over the other operation. You can think of distributing as copying the operation to each of the operands of the other operation.

Example

The operation of multiplication is distributive with respect to addition. For example:

2 \cdot (3 + 4) = (2 \cdot 3) + (2 \cdot 4) = 14

The operation of division is not distributive with respect to addition. For example:

2 \div (3 + 4) \neq (2 \div 3) + (2 \div 4)

Left and Right Distributivity

An operation $*$ is left distributive if for all $a, b, c \in S$ the operation is distributive with respect to $+$ when the operation is on the left:

a * (b + c) = (a * b) + (a * c)

not necessarily for the right:

(b + c) * a = (b * a) + (c * a)

Existential Clauses

There are also operations that define existential clauses, i.e the operation requires the existence of a special element that satisfies a certain property.

Identity

Inverse

Categories of Algebraic Structures

Magma

Semigroups

also called half-group, just need to be associative

Commutative Semigroups

If it is also commutative, it is called a commutative semigroup or abelian semigroup. most operations that are also commutative are either called so or abelian.

Monoids

Inbetween semigroups and groups, Is a semigroup with an identity element. Therfore obeying all the properties of a group except for the inverse property.

Why are these useful especially in computer science?

Groups

CAIN properties, c is closure

accosiativy, identity, inverse

int with +, non-zero real numbers with multiplication

Commutative Groups

If it is group and commutative, also called a abelian group

Cyclic Groups

is a group that is generated by a single element, composition table to see if it is cyclic

Rings

Should already be a abelian group under both operations?

two operations, + and *, called addition and multiplication (don't have to be the same as the normal ones). needs closure under multiplication and associative.

commutative group under addition
associative under multiplication

There is a multiplicative Identity??? unclear

is distributive over +

Commutative Rings

if the multiplication operation is commutative

Fields

integral domain with multiplicative inverse

also somehow relates to vector spaces

Integral Domains

commutative ring with multiplicative identity that also has no zero divisors

Finite Fields

also known as Galois fields if set S is finite. important in cryptography because of integers modulo a prime number

Modules

generalization of vector spaces, but instead of scalars being in a field, they are in a ring??

Partial Orders Homomorphisms