A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$ (for instance, every commutative semigroup is duo, and so is every group). On the other hand, we say that $S$ is a left/right cancellative semigroup if, for each $a \in S$, left/right multiplication by $a$ is an injective function on $S$. Then, $S$ is called cancellative if it is both left and right cancellative.

I recently asked (here) if a cancellative semigroup embeds into a group. Pace Nielsen promptly answered the question in the affirmative (here). In a comment to Pace's answer (here), I then raised the following:

Q. If a duo semigroup is left/right cancellative, is it cancellative? Any reference?

There is a certain abundance of non-commutative, cancellative duo semigroups in nature (see here for a short list). Yet, one-sided cancellative, duo semigroups appears to be 'rare' (of any exists). Among other things, note that every finite left/right cancellative duo semigroup is a group.

A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$; for instance, every commutative semigroup is duo, and so is every group. On the other hand, we say that $S$ is a left (resp., right) cancellative semigroup if, for each $a \in S$, left (resp., right) multiplication by $a$ is an injective function on $S$. Then, we call $S$ cancellative if it is both left and right cancellative.

I recently asked (here) if a cancellative semigroup embeds into a group. Pace Nielsen promptly answered the question in the affirmative (here). In a comment to Pace's answer (here), I then raised the following question:

Q. If a duo semigroup is left/right cancellative, is it cancellative? Any reference?

There is a certain abundance of non-commutative, cancellative duo semigroups "in nature" (see here for a short list). Yet, one-sided cancellative, duo semigroups appear to be "rare" (if any exist). Among other things, note that every finite left/right cancellative duo semigroup is a group.

A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$ (for instance, every commutative semigroup is duo, and so is every group). On the other hand, we say that $S$ is a left/right cancellative semigroup if, for each $a \in S$, left/right multiplication by $a$ is an injective function on $S$. Then, $S$ is called cancellative if it is both left and right cancellative.

I recently asked (here) if a cancellative semigroup embeds into a group. Pace Nielsen has promptly answered the question in the affirmative (here). In a comment to Pace's answer (here), I then raised the following:

Q. If a duo semigroup is left/right cancellative, is it cancellative? Any reference?

Of course, the answer is yes in the commutative setting. On the other hand, there is a certain abundance of non-commutative, cancellative duo semigroups in nature (see here for a short list). Yet, the existence of one-sided cancellative, duo semigroups is baffling to me. In particular, note that every finite left/right cancellative duo semigroup is a group (and hence cancellative).

A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$ (for instance, every commutative semigroup is duo, and so is every group). On the other hand, we say that $S$ is a left/right cancellative semigroup if, for each $a \in S$, left/right multiplication by $a$ is an injective function on $S$. Then, $S$ is called cancellative if it is both left and right cancellative.

I recently asked (here) if a cancellative semigroup embeds into a group. Pace Nielsen promptly answered the question in the affirmative (here). In a comment to Pace's answer (here), I then raised the following:

Q. If a duo semigroup is left/right cancellative, is it cancellative? Any reference?

There is a certain abundance of non-commutative, cancellative duo semigroups in nature (see here for a short list). Yet, one-sided cancellative, duo semigroups appears to be 'rare' (of any exists). Among other things, note that every finite left/right cancellative duo semigroup is a group.