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^#