Conventionally the domain of discourse specifies the set of objects are we addressing. Within mathematical logic, the domain is specified outside of the axioms and transformation rules, as a Limitation on the type of objects that the transformations apply to. (Iconic Arithmetic Volume I, p. 216)

For example, every number has a successor,

∀n n ∈ ℕ ⇒ n + 1 ∈ ℕ.

The universal quantification ∀n asserts we are referring to all of the domain. The double struck ℕ indicates that the rule applies to the domain of natural numbers only. ℕ defines what we mean by “every number”. The set membership statement n ∈ ℕ, declares that whatever the variable n indexes belongs to the set of natural numbers. The rule says that for any natural number, there is a natural number that is one unit larger. The final set membership statement, n + 1 ∈ ℕ, declares that the successor number n + 1 is also a member of the set of natural numbers. Not only does the rule define a valid transformation (we can add one to any number), it also generates all admissible numbers. Combined with other rules, it contributes to the definition of our Concept of Natural Numbers.

⇒ Closure

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Servers are allocated IP addresses which then must be associated with a domain name which one leases for a few years at a time. A domain name registrar is a company that manages the reservation of Internet domain names. wikipedia

A domain "owner" can establish unlimited subdomains by appropriately configuring DNS.

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See Domain-driven design (DDD). post