Does there exist a reversible monoid that fails to be Dedekind-finite?

Question

Call a ring with unity

• reversible iff $xy = 0$ implies $yx = 0$.
• Dedekind-finite iff $xy = 1$ implies $yx = 1$.

It is proved here that every reversible ring is Dedekind-finite.

Now clearly, the above definitions make sense for an arbitrary monoid with an absorbing element $0$ satisfying $x0 = 0$ and $0x = 0$. Call such a structure a monoid with zero.

Does there exist a reversible monoid with zero that fails to be Dedekind-finite?

Answer 1

Take the bicyclic monoid B and add a zero. The resulting monoid is a reversible monoid with zero that fails to be Dedekind-finite.