#algebraic structures

[eot-tap1b-opt]

Properties of Number Systems/Algebraic Structures

Numbers have some properties that are really obvious. So obvious that we may learn them without ever realizing it. For example $a+b=b+a$. This is called the commutative property of addition and it also applies to multiplication. Similarly $(a+b)+c=a+(b+c)$ is the associative property. It might seem silly to bother thinking about these things because we already intuitively understand them. Things get interesting however when we look at algebras that have these properties, but are not like the ordinary numbers.

If an algebra satisfies certain properties then we call it a kind of algebraic structure. We'll look at the following algebraic structures:

Monoids

Commutative Monoids

Groups

Commutative (aka abelian) Groups

Rings

Commutative Rings

Fields

Monoids

A monoid is a triple $(S,\cdot,e)$ where $e\in{}S$ is called the identity, $\cdot:S\times{}S\rightarrow{}S$ and $S$ is an arbitrary set. There are two rules for a triple to be a monoid:

Associativity: For any $x$, $y$ and $z$ in $S$ we must have $(x\cdot{}y)\cdot{}z=x\cdot{}(y\cdot{}z)$

Identity: For any $x$ in $S$ we must have $x\cdot{}e=e\cdot{}x=x$

Example

Let $S$ be the set of all lists of integers. A list is just a finite sequence, for example $[1,2,3,1]$. Let $x\cdot{}y$ be the result of concatenating x and y, for example $[1,2]\cdot{}[3,1]=[1,2,3,1]$. Let $e$ be the list of length 0 $[]$. This forms a monoid. Notice that it is not commutative: $[1]\cdot{}[2]\neq{}[2]\cdot{}[1]$

Commutative Monoids

A commutative monoid is a monoid $(S,\cdot,e)$ with the additional property that for all $x$ and $y$ $x\cdot{}y=y\cdot{}x$.

Two Examples

Let $S$ be the natural numbers, $\cdot$ be addition and $e$ be $0$

Let $S$ be the non-zero natural numbers, $\cdot$ be multiplication and $e$ be $1$

Commutitive Groups

A group is a monoid $(S,\cdot,e)$ with the additional property that for every element $x\in{}S$ there is another element, denoted $x^{-1}$, also in $S$ for which $x\cdot{}x^{-1}=e$. If it's also true that $x\cdot{}y=y\cdot{}x$ then we say it's an abelian or commutative group.

Examples

Let $S$ be $\mathbb{Z}$, $\cdot$ be addition and $e$ be 0.

Let $S$ be $\mathbb{Q}\setminus\{0\}$, $\cdot$ be multiplication and $e$ be $1$

Non-Example

Let $S$ be $\mathbb{Z}\setminus\{0\}$, $\cdot$ be multiplication and $e$ be 1. $1$ and $-1$ have inverses, but there's no integer $n$ such that $2n=1$.

Non-Commutative Groups

Consider a 6-sided die. Depending on how we hold it there will be one number on the top and another on the front. Knowing these two numbers is enough the figure out where all the others lie. There are 24 such combinations (6 possibilities for the top $\times$ 4 possibilites for the side). We can rotate the die $90^{o}$ along any axis to obtain a new orientation. For each orientation there is a way of rotating the die from its starting position to that orientation. Let the set of orientations be $S$. Let $x\cdot{}y$ be the orientation formed by first applying the rotations that give x from the starting position, then the rotations for y. $e$ is the identity, when we apply no rotations at all, or go $360^{o}$ around an axis. This is called the symmetry group of the cube.

Is this a group? Let's check.

The composition of two rotations is another rotation

$e$ is obviously an identity

For every rotation we can just do the opposite to get the inverse

It doesn't matter if we first rotate about z and then give it to someone to rotate about x and y, or if we rotate and z and x and then give it to someone to rotate about y. Either way we get the same final orientation

So this is a group. To see that it's not commutative check out the linked video. We have 4 transformations:

$a$: $90^{o}$ counter-clockwise about the z-axis

$b$: $90^{o}$ clockwise about the y-axis

$a^{-1}$: $90^{o}$ clockwise about the z-axis

$b^{-1}$: $90^{o}$ counter-clockwise about the y-axis

If we apply them in the order $abb^{-1}a^{-1}$ then obviously they cancel out and we're left with $e$. In the video we apply them in the order $aba^{-1}b^{1}$ and are left with a new orientation.

Rings

A ring is defined by a 5-tuple $(S,+,\times,0,1)$ and is basically a combination of a group and a monoid. It has following rules:

$+$ and $\times$ are functions from $S\times{}S$ to $S$

$(S,+,0)$ is a commutative group

$(S\setminus\{0\},\times{},1)$ is a monoid

For all $a$, $b$ and $c$ in $S$: $a\times(b+c)=a\times{}b+a\times{}c$ and $(b+c)\times{}a=b\times{}a+c\times{}a$

The integers, rationals, reals and complex numbers all form rings under normal addition and multiplication.

Commutative ring

In a ring addition $(+)$ always has to be commutative. If we have the additional property that multiplication $(\times)$ is commutative then we say it is a commutative ring.

Field

A field $(S,+,\times,0,1)$ is a commutative ring that has a multiplicative inverse for each number except for zero. That is to say: for all $x\in{}S\setminus\{0\}\exists{}x^{-1}\in{}S:xx^{-1}=x^{-1}x=1$. The rationals, reals and complex numbers are fields but the integers are not.

Those are the most important algebraic structures but there are many others. Some combinations of properties can't exist (try defining a field where $0$ has an inverse), while others can be quite exotic. Next time we'll look at some interesting fields.

[that-average-fangirl]

This is my book stack & my roommates shutter at the sight of it

[powermodules]

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. On the other hand, any monoid can be understood as a special sort of category, and so can any preorder. In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships. This is the central idea of category theory, a branch of mathematics which seeks to generalize all of mathematics in terms of objects and arrows, independent of what the objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. For more extensive motivational background and historical notes, see category theory and the list of category theory topics.

Two categories are the same if they have the same collection of objects, the same collection of arrows, and the same associative method of composing any pair of arrows. Two categories may also be considered "equivalent" for purposes of category theory, even if they are not precisely the same.