I am reading this paper, and in section 4, they define the "monoïde cycliste" to be the quotient of the free monoid on two generators, $\{\alpha,\beta\}^*$, by the relations $\alpha\alpha\beta \sim \alpha \beta \alpha$ and $\beta \alpha\beta \sim \alpha\beta\beta$.
I am unsure how to translate "monoïde cycliste." I was first tempted to say "cyclic monoid," but at the end of the wikipedia page, we see that a cyclic monoid is the analogue of a cyclic group, which happens to be "groupe cyclique," not "groupe cycliste." Does anyone know a good way to translate "cycliste"? Maybe "cyclist monoid"? Thank you in advance.
Welcome to the humour of (the late) Alain Lascoux, and Marco Schützenberger. In French a cycliste is one who rides a bicycle, and as an adjective it means in relation to (the sport of) riding bicycles. Since that monoid has two generators, they no doubt found the allusion to bicycles appropriate. It would seem to me there is no really pressing need to give the two-letter instance of the plactic monoid a special name, but as a translation "cyclist monoid" is as good as any. As an alternative option I could propose "bicyclic monoid", although that could possibly be confused with the free monoid with two generators.