Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

Semigroups [and axioms]

Despite their simpler definition, semigroups have been much less studied than groups, and there have for example been about 7 times fewer mathematical publications about them (and another 7 times fewer about monoids). Semigroups were defined by Jean-Armand de Séguier in 1904, and beginning in the late 1920s a variety of algebraic results about them were found. Since the 1940s they have showed up sporadically in various areas of mathematics—notably in connection with evolution processes, finite automata and category theory.

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