Negative domains. Negative integers, Z#
A set A ⊆ Z is called a negative domain ⇐:⇒ the following two conditions are satisfied:0
- 1 ∉ A
- ∀x ∈ A x-1 ∈ A
The union of any family of negative domains is a negative domain. In particular the union Z# of all negative domains is called the set of negative integers. It contains every other negative domain.
The following theorem is straightforward:
THEOREM ?? A set of integers B ⊆ Z is a negative domain ⇔ A := Z \ B is a natural domain.
Since N is a natural domain, and 0 ∉ N, it follows that 0 ∈ Z# — indeed,
0 ∈ Z \ N ⊆ Z#