See also:

  • discrete math
  • category theory
  • linear algebra
  • nonlinear algebra

Abstract Algebra

Elementary

An informal term, but you know. High school stuff and less. Quadratic Equation Manipulating both sides

Monoids

class Monoid (A : Type) where
  prod : A -> A -> A
  e : A
  assoc : forall x y z, prod (prod x y) z = prod x (prod y z)
  
def main : IO Unit := return ()

Groups

I don’t know why, but this is still the one that makes me the most comfortable

  • Representatations
  • Permutations
  • Symettries
  • Lie Groups

https://en.wikipedia.org/wiki/Computational_group_theory GAP Magma

Semirings

Fun with semirings: a functional pearl on the abuse of linear algebra

Semiring provenance in datalog

Rings

  • See nonlinear algebra

Modules

Fields

finite fields/galois field. It’s kind of remarkable finite fields even exist. A well behaving notion of number that includes division but is nevertheless finite

Vector Spaces

  • See linear algebra

Lattices

Galois connections Fixed point theorems

Boolean Algebra

Heyting Algebra

Kleene

  • See automata

Algebra over a field

https://en.wikipedia.org/wiki/Algebra_over_a_field

Geometrical Algebra clifford algebra exterior algebra

Matroids

https://www.cambridge.org/us/academic/subjects/computer-science/algorithmics-complexity-computer-algebra-and-computational-g/computational-oriented-matroids-equivalence-classes-matrices-within-natural-framework?format=HB&isbn=9780521849302

Tropical Algebra

Universal Algebra

Homomorphisms

https://en.wikipedia.org/wiki/Homomorphism

At a certain point of sophistication, this becomes all you care about it seems. Exactness