ABSTRACT ALGEBRA IN 3 HOURS!Ashwin Rao

Meant to be a quick preparation for learning Category Theory

Overview of Preliminaries• Set: unordered and unique elements

• Cartesian Product of Sets

• Relation: A subset of a cartesian product

• Reflexive, Symmetric, Transitive Relation on a set ó Equivalence Classes (Partition)

• Partially Ordered Set: Reflexive, Anti-symmetric and Transitive

• Function: Just a relation on A x B with every a in A mapped to a single b in B

• Domain, Codomain, Range, Injective, Surjective, Bijective functions

• Inverse and Composition of functions

Semigroup• A set with an operation (*) under which the set is closed, along with associativity.

• Associativity: a * (b * c) = (a * b) * c

• Commutativity a * b = b * a is fairly common, but not part of semigroup definition.

• Canonical Example: Positive Integers Z+ with operation as + or *

• Funky Example: Integers with Min or Max operation.

• Example: Free semigroup of an alphabet (List[T] except empty list, with concat)

• Or, List[T] of length n, for any n in Z+

• Eg: Set of Functions f : X -> X with composition (think “shrinking” functions)

• Sub-Semigroup example: nZ+, for n in Z+

• Semigroup homomorphism (structure-preserving) f: G -> H : f(a *G b) = f(a) *H f(b)

Monoid• Semigroup together with an identity element (call it “1”)

• Canonical Example: Natural numbers N with + as * , 0 as 1

• or, Z+ with * as * and 1 as 1

• Example: Free monoid of an alphabet (List[T] with concat)

• Or, List[T] of length n, for any n in N.

• Eg: {True, False} with AND as *, True as 1 (or with OR as *, False as 1)

• Eg: All subsets of a set S with Union as *, Empty as 1 (or Intersect as *, S as 1)

• Note: Cartesian product of monoids is a monoid

• Note: All functions from a set to a monoid form a monoid (pointwise operation)

• Eg: All Functions f: X -> X for any set X, with composition as * and identity function as 1

Monoid (continued)• Submonoid example: nN

• Monoid homomorphism f: G -> H : f(a *G b) = f(a) *H f(b) and f(1G) = 1H

• Example: f(x) = 2x from (N,+,0) to (N,*,1)

• Isomorphism is when we have homomorphisms f: G -> H and g: H -> G such that g . f = idG and f . g = idH

• Isomorphism means the two monoids are “basically the same”

• Kernel(f) = {a in G | f(a) = 1H} is a monoid

• Isomorphism can also be defined as a homomorphism f with Kernel(f) = {1G}

• Note: The f(x) = 2x example is an isomorphism

Group• Monoid together with an inverse a-1 for every a such that a * a-1 = a-1 * a = 1

• Canonical Example: Z

• Eg: Bijective functions f : X -> X for any set X with {func composition, identity func, inverse func}

• Great Example: All Permutations of a finite set of size n (refered to as Sn)

• Eg: n-th complex root of unity zn and its powers (zn is called the generator of the group)

• Example of subgroup: nZ for any n in Z+

• Homomorphism f: G -> H: f(a *G b) = f(a) *H f(b), f(1G) = f(1H), f(a-1) = f(a)-1, eg: Z -> nZ

• Coseta,H for any a in G and any subgroup H if defined as: {a + h: h in H}

• Quotient Group: G/H is a group consisting of all the cosets of H (H becomes identity element)

• Canonical Example of Quotient Group: Z / nZ = Zn (Integers modulo n for any n in Z+)

• Isomorphism is same as defined for a monoid (isomorphism means “basically the same group“)

• First Isomorphism Theorem: Homomorphism f: G -> H, Kernel(f) is a subgroup of G, Range(f) is a subgroup of H, G/Kernel(f) is isomorphic to Range(f)

Semiring and Ring• Semiring has two monoid operations (*,1) and (+,0) with a * (b + c) = (a * b) + (a * c), (a + b) * c = (a * c) + (b * c), and 0 * a = a * 0 = 0. Moreover, + is commutative.

• Canonical Example: N

• Ring is a semiring with + operation having an inverse (i.e., a group under +)

• Ring Homomorphism means homomorphism under both + and *

• Canonical Example: Z

• Another Canonical Example: Polynomials over R

• Ideal I is a subset of Ring R s.t. for any x, y in I and r in R, x + y and r * x are in I

• Canonical Example of Ideal: nZ

• R / I is a ring (Quotient Ring) consisting of all the cosets of I s.t. (a+I)+(b+I) = (a+b)+I and (a+I)*(b+I) = (ab)+I

Field• Field is a ring with an inverse for *, and * commutative.

• Canonical Example: Rational Numbers Q or Real Numbers R

• Finite Field Example: Zp for any prime p

• Every finite field is isomorphic to the set of polynomials over the finite field Zpmodulo an irreducible polynomial (over Zp)

• Hence, finite fields are of size pr (r is the degree of the irreducible polynomial)

Vector Space and Linear Map• Vector Space V (associated with scalar Field F) is a commutative group under vector addition, together with scalar multiplication, and the following properties:o a(bv) = (ab)vo 1(v) = vo a(u+v) = au + avo (a+b)v = av + bv

• Canonical Example: Rn

• Eg: Complex numbers and other field extensions

• Eg: Functions from a set X to a field F (pointwise addition and pointwise scalar multiplication)

• Linear Map f: V -> W has property f(v+w) = f(v) + f(w) and f(a.x) = a.f(x)

• Canonical Example: m by n Matrix M: Rn -> Rm

• Linear maps V -> W forms a vector space L(V,W)

• Linear maps V -> F (F the scalar Field) is called the Dual Vector Space V*

Fundamental Theorem of Linear Algebra• Consider a linear map expressed as a m x n matrix M : Rn -> Rm

• Column Space (Range): Subspace of Rm consisting of all Mx (over all x in Rn)

• Row Space (CoRange): Subspace of Rn consisting of all MTy (over all y in Rm)

• Kernel Space: Subspace of Rn mapping (through M) to 0 in Rm

• CoKernel Space: Subspace of Rm mapping (through MT) to 0 in Rn

• Rank r is defined as the dimension of Column Space(= Dimension of Row Space)

• Kernel Space is orthogonal to Row Space and has rank n – r (a.k.a. Nullity)

• CoKernel Space is orthogonal to Column Space and has rank m – r (a.k.a. CoRank)

• More generally, we know from the First Isomorphism Theorem (on Groups) that the Kernel Quotient (i.e., Row Space) and Range (i.e., Column Space) are isomorphic.