Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Groups and semigroups [and operator systems]

With k possible values for each variable, the forms of operators allowed by axiom systems for group theory and semigroup theory correspond to multiplication tables for groups and semigroups with k elements. Note that the first group that is not commutative (Abelian) is the group S3 with k = 6 elements. The total number of commutative groups with k elements is just

Apply[Times, Map[PartitionsP[Last[#]] &, FactorInteger[k]]]

(Relabelling of elements makes the number of possible operator forms up to k! times larger.) (See also pages 945, 1153 and 1173.)

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]