View Source Mathematical Background

𝕋erminology

The pair of two elements is called a Slot if the following conditions are met

  • each element is either nil or an instance of a datetime with defined timezone
  • if both elements are datetimes, the first element does not superseed the second one
  • if either element is nil, the Slot is called open, if both are nil, it’s called identity

Slot id denoted [from → to]. Let’s define a binary union operation on slots, denoted ∪. Slots do not form a group with ∪, but sorted sets of slots of arbitrary length having no joint slots (denoted 𝕥 or more verbose 𝕥[[from₁, to₁], [from₂, to₂], …]) do indeed form a group, denoted 𝕋, together with a binary operation ∪ on 𝕋, such as the following group axioms are satisfied:

Associativity

∀ 𝕥₁, 𝕥₂, 𝕥₃ ∈ 𝕋, (𝕥₁ ∪ 𝕥₂) ∪ 𝕥₃ = 𝕥₁ ∪ (𝕥₂ ∪ 𝕥₃)

Identity element

∃ 𝕥₀ ∈ 𝕋 (𝕥[]) such that, for every 𝕥 in 𝕋, (𝕥 ∪ 𝕥₀) = (𝕥₀ ∪ 𝕥) = 𝕥

Inverse element

For each 𝕥 in 𝕋, there exist 𝕥¯¹ such that 𝕥 ∪ 𝕥¯¹ = 𝕥¯¹ ∪ 𝕥 = 𝕥₀

That said, slots form an Abelian group with union binary operation and empty set as identity element.

Slots Semigroup

Slots themselves form a semigroup with a binary union operation, an identity element [nil → nil], without inverse.

Binary Operation

∀ 𝕥₁ = 𝕥[[from₁ → to₁]], 𝕥₂ = 𝕥[[from₂ → to₂]] ∈ 𝕋, 𝕥₁ ∪ 𝕥₂ is defined as

  • 𝕥[[from₁ → to₁], [from₂ → to₂]] if to₁ < from₂
  • 𝕥[[from₂ → to₂], [from₁ → to₁]] if to₂ < from₁
  • 𝕥[[min(from₁, from₂) → max(to₁ → to₂)]] otherwise

nil is considered to be less than any datetime and greater than any datetime, thus 𝕥[[nil → to₁]] ∪ 𝕥[[from₂ → to₂]] would be either 𝕥[[nil → max(to₁, to₂)]] if to₁ > from₂ or 𝕥[[nil → to₁], [from₂ → to₂]] otherwise.

Mergeability

Once 𝕋 is a group, each two elements of it might be merged. Even if they are infinite. That’s why Stream implementation of 𝕋 exists.