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A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a product $g=ab$ for some $a\in A$ and $b\in B$.

My question is: Does there exist a u.p.-group (or even u.p.-semigroup) for which the corresponding property with three sets $A,B,C$ doesn't hold? Meaning, must it be the case that there is an element $g\in A\cdot B\cdot C$ which is uniquely represented in the form $g=abc$ for some $a\in A,b\in B,c\in C$?

Necessarily, such a group would not be orderable, so such an example might be difficult to find. Then again, maybe some of the recently constructed examples would suffice?

Edited to add: I found an example of a u.p.-semigroup which does not have this triple product property, which lends credence to the belief that the same is true for groups.

A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a product $g=ab$ for some $a\in A$ and $b\in B$.

My question is: Does there exist a u.p.-group (or even u.p.-semigroup) for which the corresponding property with three subsets $A,B,C$ doesn't hold? Meaning, must it be the case that there is an element $g\in A\cdot B\cdot C$ which is uniquely represented in the form $g=abc$ for some $a\in A,b\in B,c\in C$?

Necessarily, such a group would not be (two-sided) orderable, so such an example might be difficult to find. Then again, maybe some of the recently constructed examples would suffice?

Edited to add (July '16): I found an example of a u.p.-semigroup which does not have this triple product property, which lends credence to the belief that the same is true for groups.

A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a product $g=ab$ for some $a\in A$ and $b\in B$.

My question is: Does there exist a u.p.-group (or even u.p.-semigroup) for which the corresponding property with three sets $A,B,C$ doesn't hold? Meaning, must it be the case that there is an element $g\in A\cdot B\cdot C$ which is uniquely represented in the form $g=abc$ for some $a\in A,b\in B,c\in C$?

Necessarily, such a group would not be orderable, so such an example might be difficult to find. Then again, maybe some of the recently constructed examples would suffice?

A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a product $g=ab$ for some $a\in A$ and $b\in B$.

My question is: Does there exist a u.p.-group (or even u.p.-semigroup) for which the corresponding property with three sets $A,B,C$ doesn't hold? Meaning, must it be the case that there is an element $g\in A\cdot B\cdot C$ which is uniquely represented in the form $g=abc$ for some $a\in A,b\in B,c\in C$?

Necessarily, such a group would not be orderable, so such an example might be difficult to find. Then again, maybe some of the recently constructed examples would suffice?

Edited to add: I found an example of a u.p.-semigroup which does not have this triple product property, which lends credence to the belief that the same is true for groups.

A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a product $g=ab$ for some $a\in A$ and $b\in B$.

My question is: Does there exist a u.p.-group (or even u.p.-semigroup) for which the corresponding property with three sets $A,B,C$ doesn't hold? Meaning, must it be the case that there is an element $g\in A\cdot B\cdot C$ which is uniquely represented in the form $g=abc$ for some $a\in A,b\in B,c\in C$?

Necessarily, such a group would not be orderable, so such an example might be difficult to find.

A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a product $g=ab$ for some $a\in A$ and $b\in B$.

My question is: Does there exist a u.p.-group (or even u.p.-semigroup) for which the corresponding property with three sets $A,B,C$ doesn't hold? Meaning, must it be the case that there is an element $g\in A\cdot B\cdot C$ which is uniquely represented in the form $g=abc$ for some $a\in A,b\in B,c\in C$?

Necessarily, such a group would not be orderable, so such an example might be difficult to find. Then again, maybe some of the recently constructed examples would suffice?