1. ABSTRACT ALGEBRA IN 3 HOURS!
Ashwin Rao
Meant to be a quick preparation
for learning Category Theory

2. 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

3. 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)

4. 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

5. 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

6. 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)

7. 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

8. 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 Zp
modulo an irreducible polynomial (over Zp)
• Hence, finite fields are of size pr (r is the degree of the irreducible polynomial)

9. 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)v
o 1(v) = v
o a(u+v) = au + av
o (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*

10. 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.