Reading part three of Category Theory Illustrated by Jencel P:

• Previous reading is 20251009175106
• What are monoids?
  • A monoid is made up of three things: A set of elements called the underlying set, an operation called the monoid operation, and an identity element.
  • The monoid operation is a rule for combining two items in the underlying set to get another item in the underlying set. The monoid operation is associative.
  • The identity element, when combined with any other element A gives back that same element A.
  • A semigroup is made up of two things: A set of elements called the underlying set and an operation called the monoid operation.
• Basic monoids
  • In general, monoids are denoted by specifying the set and the operation enclosed in angle brackets.
  • The set of natural numbers ℕ and the addition operation is known as the monoid ⟨ℕ, +⟩ .
  • Task 2:
    • ⟨ℤ, +⟩ is a monoid, with 0 as identity
    • − is not a monoid because it is not associative
    • ⟨ℕ, ×⟩ is a monoid, with 1 as identity
    • ÷ is not a monoid because it is not associative
  • Task 5:
    • The identity element for ∧ is true and the identity for ∨ is false
• Other monoid-like objects
  • Commutative monoids (also known as abelian monoids) monoids with monoid operations that are commutative. For example, ⟨ℤ, +⟩ is a commutative monoid.
  • A group is a monoid that has an underlying set for which each element can be paired with another element in that set that, when combined together with the monoid's monoid operation, gives you the identity element.
    • Monoids can be viewed as a way to model the effect of applying a set of associative actions.
    • Groups can be viewed as a way to model the effects of actions which are also reversible.
    • For example, ⟨ℤ, +⟩ is a group.
• Symmetry groups
  • Consider a set of all rotations ℝ in the monoid ⟨ℝ, +⟩. The rotations themselves make up the underlying set, with the zero degree rotation as the identity and the successive application of two rotations as the monoid operation. This monoid is a group.
  • For triangles, every rotation in this group can be reduced to the repeated application of the 120-degree rotation and for octogans, every rotation in this group can be reduced to the repeated application of the 45-degree rotation. This smaller rotation from which you can make all of the other bigger rotations in the group could be considered the "main rotation".
  • Groups that have such "main rotation" are called cyclic groups. The "main rotation" is called the group’s generator.
  • The the set of natural numbers ℕ is a cyclic monoid under addition with +1 as its generator and the set of integers ℤ is a cyclic group under addition with either +1 or −1 as its generator.
  • They mention an "inverse wall" but it is not mentioned or defined elsewhere in the book. I think they mean you can always use a generator's inverse to generate all objects in a group, since groups are invertible.
  • The set of integers ℤ is an infinite cyclic group.
  • The group of rotations for a triangle and the group of integers under addition with modulo 3 have group isomorphism.
  • All cyclic groups are commutative.