Regular cardinal
An infinite
cardinal number κ is called
regular if cf(κ) = κ, where cf is the
cofinality operation. This says that κ cannot be expressed as the
union (
supremum) of a collection of less than κ smaller cardinals. If we demand a regular cardinal be also a
limit cardinal, we get an
inaccessible cardinal.
Cardinals which are not regular are called singular (the existence of singular cardinals requires the Axiom of replacement.)