TITLE: Examples of left reversible semigroups
QUESTION [8 upvotes]: I am looking for concrete examples of cancellative, left reversible semigroups. Left reversible semigroups are also called "Ore semigroups". See this wikipedia page for the definition of a left reversible semigroup. Of course, commutative semigroups are automatically left reversible, and I am looking for non-commutative examples.
Please also mention if these semigroups arise in an interesting setting.

REPLY [3 votes]: The semigroup $S= \langle a,b\mid a^2=b^2\rangle$ is two-sided reversible.