From Semigroup To Group, Through Power Notation


If we consider the exponential notation used in abstract algebra, we get a nice mnemonic for semigroups, monoids, and groups:

  • Semigroup: Only positive powers are defined
  • Monoid: The identity element makes \(a^0 = e\) defined, so non-negative powers are defined
  • Group: inverses define negative powers, so all integer powers are defined

By remembering the completion of the natural numbers into the integers, we can remember what exactly defines semigroups, monoids, and groups (I know I mix them up all the time).

Related Posts

Random stuff

Lossless Data Compression with Neural Networks by Fabrice Bellard

Downscaling Numerical Weather Models With GANs (My CI 2019 Paper)

Learning Differential Forms and Questions

PyTorch Lightning is worth using

Feynman Lectures, Chapter 2

Feynman Lectures 1

Random Julia Thoughts

Death Still Sucks

How I feel about ebooks