math-structures.org

\href{http://math.chapman.edu/~jipsen/structures/doku.php}{Classes of math structures}

Algebra

Algebra is a lot like arithmetic. In arithmetic the only unknown thing is the answer to the problem. Algebra follows the rules of arithmetic.

Algebra uses operations of arithmetic: $+$, $-$, $*$, $/$ in equations: $1 + 2 = …$

Algebra adds the element of “unknown”, typically a letter $x$ which is a placeholder for some unknown value in the equation: $1 + 2 = x$

Goals of algebra: what is the value of $x$? I.e. it’s about “solving equations”

Abstract Algebra, aka Modern Algebra

Algebraic structures

\href{https://en.wikipedia.org/wiki/Algebraic_structure}{Algebraic structure}

Group-like structures

Group is a pair $(G, ⋅)$:
$G$ is an underlying body of an algebraic structure. Often it’s a set. $⋅$ is a binary operation. The element inversion $\blank^-1$ is apparently not always being consider to be an operation.

Group-like-Structure	Closed	Associative	Identity	Invertible	Commutative
Semigroupoid	-	Y	-	-	-
Small category	-	Y	Y	-	-
Groupoid	-	Y	Y	Y	-
Magma	Y	-	-	-	-
Quasigroup	Y	-	-	Y	-
Loop	Y	-	Y	Y	-
Semigroup	Y	Y	-	-	-
Inverse semigroup	Y	Y	-	Y	-
Monoid	Y	Y	Y	-	-
Group	Y	Y	Y	Y	-
Abelian group	Y	Y	Y	Y	Y

Closed / Closure property: all possible results of $a ⋅ b$ are in the underlying body
Associative, ie group with parentheses as you like, ie. no priority: $a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c$ \ Commutative, ie. order doesn’t matter: $a ⋅ b = b ⋅ a$

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Eg. for groups (which are associative, but not commutative): the priority of operations doesn’t exist bu the order of operands is important

Semigroup / Halbgruppe:

Semigroup is tuple $(G, ⋅)$ such that:
$⋅$ is closed / closure: $a ⋅ b$ must be also in $G$ \ Not required: $⋅$ is commutative: $x ⋅ y = y ⋅ x$ \ $⋅$ is associative: $(x ⋅ y) ⋅ z = x ⋅ (y ⋅ z)$ \ Not required: $e$ is neutral element (identity) \ Not required: $\blank\inv$ inversion (dual obj): $x ⋅ x\inv = x\inv ⋅ x = e$

Monoid

Monoid is a triple $(G, ⋅, e)$ such that:
A group without inverse elements. \ $⋅$ is closed / closure: $a ⋅ b$ must be also in $G$ \ $⋅$ is associative: $(x ⋅ y) ⋅ z = x ⋅ (y ⋅ z)$ \ $e$ is neutral element (identity) \ Not required: $\blank\inv$ inversion (dual obj): $x ⋅ x\inv = x\inv ⋅ x = e$

Group

Group is a triple $(G, ⋅, e)$ such that:
A category w/ one object => every morphism is an isomorphism \ $⋅$ is closed / closure: $∀ a,b in G: a ⋅ b$ must be also in $G$ \ Not required: $⋅$ is commutative: $x ⋅ y = y ⋅ x$ e.g. matrix multiplication is not a commutative operation \ $⋅$ is associative: $(x ⋅ y) ⋅ z = x ⋅ (y ⋅ z)$ \ $e$ is neutral element (identity) \ $\blank\inv$ inversion (dual obj): $x ⋅ x\inv = x\inv ⋅ x = e$ \

A category of many morphisms and only one object. All the morphisms are invertible.

The pieces of the structure on $G$ can bee seen as maps of sets:
$⋅ : G² → G$ \ $\blank\inv : G¹ → G$ \ $e : G⁰ → G$ \

$G⁰$ is the one-element set $1$ and we are using the observation that a map $1 → G$ of sets is essentially the same thing as an element of $G$.

Normal Subgroups, Quotient (a.k.a Factor) Groups

Modular arithmetic done in the group theory by normal subgroups and quotient groups.
Congruence Classes $a ≡ b \pmod {n}$: $a$ is congruent to $b$ modulo $n$. E.g. Integers under addition $+$ form a Group:\ $0 + 3 \ZZ = \{… -6, -3, 0, 3, 6, …\}$ - Normal Subgroup \ $1 + 3 \ZZ = \{… -6, -3, 0, 3, 6, …\}$ - Coset \ $2 + 3 \ZZ = \{… -6, -3, 0, 3, 6, …\}$ - Coset \

A (Normal) Subgroup partitions a Group into (non-overlapping) Cosets. $0 + 3\ZZ$ is a Normal Subgroup and Cosets $1 + 3\ZZ$, $2 + 3\ZZ$ form a Quotient Group $\ZZ / 3\ZZ$ a.k.a. a Factor Group.
Notes:

• the Quotient Group $\ZZ / 3\ZZ$ is not a Subgroup of $\ZZ$, i.e. $\ZZ / 3\ZZ \nleq \ZZ$. It is an entirely different group.
• in general the Cosets do not always form a Group.

Cyclic Group: generated by one element.

Commutative / Abelian group

Abelian group is a triple $(G, ⋅, e)$: elements with multiplication, such that:
Formalization of symmetry concept. \ A symmetry of an object $G$ is a way of mapping $G$ to itself in a reversible or invertible manner.\ $⋅$ is closed / closure: a ⋅ b must be also in $G$ \ $⋅$ is commutative: $x ⋅ y = y ⋅ x$ \ $⋅$ is associative: $(x ⋅ y) ⋅ z = x ⋅ (y ⋅ z)$ \ $e$ is neutral element (identity) \ $\blank\inv$ inversion (dual obj): $x ⋅ x\inv = x\inv ⋅ x = e$

Ring-like structures

Semiring - Ring w/o additive inverse $− a$; Ring w/o i: R(i)ng

• $R$ is commutative i.e. abelian monoid under addition: $+$ is associative: $(a + b) + c = a + (b + c)$
  $+$ is commutative: $a + b = b + a$ \ $0$ is additive identity: $a + 0 = a$ \ Not required: additive inverse $a + (− a) = 0$, i.e. $∃ a ∈ R: − a ∉ R$
• $R$ is monoid under multiplication: $⋅$ is associative: $(a ⋅ b) ⋅ c = a ⋅ (b ⋅ c)$
  $1$ is multiplicative identity: $a ⋅ 1 = 1 ⋅ a = a$
• Multiplication is distributive with respect to addition: left distributivity: $a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c)$
  right distributivity: $(b + c) ⋅ a = (b ⋅ a) + (c ⋅ a)$

Near-ring

Commutative ring

Integral domain

Field

Field is a triple $(X, +, −)$ such that $X$ is an underlying set forming a commutative, i.e. abelian group under the operations of:
(commutative) addition: $a + b = b + a$ \ (commutative) multiplication: $a × b = b × a$, for non-zero elements of $X$, to avoid division by $0$ \ addition and multiplication are distributive: $a × (b + c) = a × b + a × c$ \

Examples:
$\CC$ Complex, $\RR$ Real and $\QQ$ Rational numbers form a Field. $\ZZ$ Integer numbers don’t form a Field. The $\ZZ$ don’t have multiplicative inverses: $$2\inv = \dfrac{1}{2} ∉ \ZZ$$

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TODO is Field a ring-like structure? I.e. does it have an associative multiplication operation $⋅$?

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See \href{https://www.johndcook.com/blog/2017/03/26/visualizing-kinds-of-rings/}{Visualizing kinds of rings}

• Euclidean domain: Every field is a Euclidean domain.
• Principal Ideal Domain: Every Euclidean domain is a principal ideal domain (PID).
• Unique Factorization Domain Every principal ideal domain is a unique factorization domain (UFD).
• Domain: Every unique factorization domain is an integral domain.
• Integral Domain: A finite integral domain is a field.

Division ring

Ring / Okruh

Ring is triple $(R, +, ⋅)$ such that:

• $R$ is commutative i.e. abelian group under addition:
  $+$ is associative: $(a + b) + c = a + (b + c)$ \ $+$ is commutative: $a + b = b + a$ \ $0$ is additive identity: $a + 0 = a$ \ existence of an additive inverse: $∀ a ∃ − a : a + (− a) = 0$
• $R$ is monoid under multiplication:
  $⋅$ is associative: $(a ⋅ b) ⋅ c = a ⋅ (b ⋅ c)$ \ $1$ is multiplicative identity: $a ⋅ 1 = 1 ⋅ a = a$
• Multiplication is distributive with respect to addition:
  left distributivity: $a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c)$ \ right distributivity: $(b + c) ⋅ a = (b ⋅ a) + (c ⋅ a)$

Ring theory

Latice-like

Lattice

Semilattice

Complemented lattice

Total order

Heyting algebra

Boolean algebra

Map of lattices

Lattice theory

Module-like

Group with operators

Vector space

Vector is quadruple $(V, +_v, F, +_f, ⋅, 1_f)$ such that:

• $V$ is an Abelian Group: $+_v$ is closed, commutative, associative, it has an identity and has invertible elements: $∀ v ∈ V ∃ v\inv ∈ V$, and it contains a collection of vectors.
• $F$ is a Field of scalars.
• Operations and their properties: commutative vector addition: $v_1 +_v v_2 = v_2 +_v v_1$
  vector scaling: $v ∈ V, f ∈ F: f ⋅ v ∈ V$ \ distributivity of $⋅$: $f ⋅ (v_1 +_v v_2) = f ⋅ v_1 +_f f ⋅ v_2$, $(f_1 +_f f_2) ⋅ v = f_1 ⋅ v +_v f_2 ⋅ v$ \ associativity of $⋅$: $f_1 ⋅ (f_2 ⋅ v) = (f_1 × \f_2) ⋅ v$ \ scaling by $1_f$: $1_f ⋅ v = v$

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Vector space:

• \href{https://youtu.be/ozwodzD5bJM}{YouTube: Socratica - What is a Vector Space?}
• any type of mathematical object that can be multiplied by numbers and added together
• collection of vectors
• Linear algebra - study of vector spaces.

Module

Module is a generalization of a Vector Space. A vector space with a ring of scalars.

Algebra-like

Algebra

Associative Non-associative Composition algebra

Lie algebra Graded Bialgebra

Intuition: Category of X

Objects(X):

Elements : Structure : Properties:

Morphisms(X):

Elements -> Elements; preserve structure

Topological spaces: Top

Objects(Top) : All Topological Spaces Morphisms(Top): Continuous Maps

Vector Spaces: Vec

Objects(Vec):

Elements : All Vector Spaces Structure: add and scale vectors Properties: adding vectors is commutative

Morphisms(Vec):

Linear transformations preserving sums and scalar multiplications - i.e. vector scaling

Posets: Pos

Objects(Pos)

Elements(Pos) : Partialy ordered sets Structure(Pos) : Order Properties(Pos): Reflexivity, Transitivity

Morphisms(Pos): TODO