  A commutative semigroup of abstract factorials is defined in the context of
the ring of integers. We study such factorials for their own sake, whether they
are or are not connected to sets of integers. Given a subset X of the positive
integers we construct a "factorial set" with which one may define a multitude
of abstract factorials on X. We study the possible equality of consecutive
factorials, a dichotomy involving the limit superior of the ratios of
consecutive factorials and we provide many examples outlining the applications
of the ensuing theory; examples dealing with prime numbers, Fibonacci numbers,
and highly composite numbers among other sets of integers. One of our results
states that given any abstract factorial the series of reciprocals of its
factorials always converges to an irrational number. Thus, for example, for any
positive integer k the series of the reciprocals of the k-th powers of the
cumulative product of the divisors of the numbers from 1 to n is irrational.
