Should I say Abelian monoïd or commutative monoïd

Question

I usually say, "Abelian group" rather than "commutative group", not sure if that's because I studied in the United states during the 1980s. But it seems people in Europe say, "commutative monoïd". I'm preparing a lecture for my computer science students where I talk about the definitions and simple examples of groups, monoïds, rings, and semirings.

The lecture covers just enough ($+\varepsilon$) to implement shortest path algorithms and generalized exponentiation. I.e., calculate $x^n$ in a monoïd, recursively by $x^n = x^{n-1} \times x$ if $n$ is odd and $x^n = (x\times x)^{\frac{n}{2}}$ if $n$ is even. And using that exponentiation step to efficiently compute matrix powers where matrix components come from tropical: semiring $(Z,min,\infty,+,0)$.

It feels strange that I always say "Abelian group", yet "commutative monoïd".

Any advice?

Answer 1

I guess "commutative" is just better. "Abelian" is widely used for groups but that should be viewed as part of an established harmless tradition, and nobody blames if you say "commutative group", except maybe your vocal cords in case of a sore throat. In the same fashion group theory comes with some obsolete terminology, such as saying "order" in lieu of "cardinal", due to the initial definition of group as permutation groups on a set, before abstract groups were introduced.

Answer 2

Although the term Abelian semigroup is used by some people, it looks like most books on semigroups use the term commutative semigroup.

This includes Clifford and Preston's classical treatise The Algebraic Theory of Semigroups, Howie's book Fundamentals of semigroup theory, Okninski's Semigroups of Matrices, my own book Varieties of formal languages, Higgins's Techniques of semigroup theory, Rhodes and Steinberg's The $q$-theory of finite semigroups, Lawson's Inverse semigroups, the theory of partial symmetries, Steinberg's more recent book Representation theory of finite monoids, and of course Redei's Theory of finitely generated commutative semigroups and Grillet's reference book Commutative semigroups.

In conclusion, I think it is safe to stick on the term commutative semigroup.