Commutative orders in semigroups

We consider commutative orders, that is, commutative semigroups having a semigroup of quotients in a local sense defined as follows. An element $a\in S$ is \em{square-cancellable} if for all $x,y\in S^{1}$ we have that $xa^2=ya^2$ implies $xa=ya$ and also $a^2 x=a^2 y$ implies $ax=ay$. It is clear that being square-cancellable is a necessary condition for an element to lie in a subgroup of an oversemigroup. In a commutative semigroup $S$, the square-cancellable elements constitute a subsemigroup $\mathcal{S}(S)$. Let $S$ be a subsemigroup $Q$. Then $S$ is a \em{left order} in $Q$ and $Q$ is a \em{semigroup of left quotients} of $S$ if every $q\in Q$ can be written as $q=a^{\#}b$ where $a\in\mathcal{S}(S)$, $b\in S$ and $a^{\#}$ is the inverse of $a$ in a subgroup of $Q$ and if, in addition, every square-cancellable element of $S$ lies in a subgroup of $Q$. \em{Right orders} and \em{semigroups of right quotients} are defined dually. If $S$ is both a left order and a right order in $Q$, then $S$ is an \em{order} in $Q$ and $Q$ is a \em{semigroup of quotients} of $S$. We remark that if a commutative semigroup is a left order in $Q$, then $Q$ is commutative so that $S$ is an order in $Q$. A given commutative order $S$ may have more that one semigroup of quotients. The semigroups of quotients of $S$ are pre-ordered by the relation $Q\geq P$ if and only if there exists an onto homomorphism $\phi:Q\to P$ which restricts to the identity on $S$. Such a $\phi$ is referred to as an $S$-\em{homomorphism}; the classes of the associated equivalence relation are the $S$-isomorphism classes of orders, giving us a partially ordered set $\mathcal{Q}(S)$. In the best case, $\mathcal{Q}(S)$ contains maximum and minimum elements. In a commutative order $S$, $\mathcal{S}(S)$ is also an order and has a maximum semigroup of quotients $R$, which is a Clifford semigroup. We investigate how much of the relation between $\mathcal{S}(S)$ and its semigroups of quotients can be lifted to $S$ and its semigroups of quotients. This is a joint work with P.N. \'Anh, V. Gould, and P.A. Grillet.