Is this a pre-ordered commutative semigroup? Motivation
I'm studying an approach to axiomatic thermodynamics based on the notion of commutative semigroup $(S,+)$ with a preorder relation $\to$ on $S$.  In other words, $S$ is non-empty set, the operation $+: S \times S \to S$ is commutative and associative, the relation $\to$ on $S$ is reflexive and transitive, and in addition there is a compatibility condition:
$$ (\forall a,b,c \in S) \qquad  a \to b \iff  a + c \to b + c $$
Question

Is this a standard (or well-known) structure?  And if so, is there an accepted term for it?

Naively I thought that it would be a pre-ordered commutative semigroup, but I googled and found very little with this name and the little I found suggests that the compatibility condition obeyed by a pre-ordered commutative semigroup is the weaker
$$ (\forall a,b,c \in S) \qquad  a \to b \implies  a + c \to b + c $$
 A: Your condition certainly was not considered by algebraists studying commutative semigroups. It implies, for example, that if the semigroup has a $0$ (i.e. $0+x=0$), then for every two $a,b$ $a\to b$ and $b\to a$. Your condition makes more sense for semigroups satisfying cancelation law: $a+c=b+c\to a=b$. Then you can embed your semigroup into a group where the two conditions are equivalent.
A: In addition to transitivity and reflexivity, to obtain the definition of a pre ordered (sometimes quasi ordered) semigroup, you need only have that the binary operation is monotone in each coordinate. That is, if $a\rightarrow b$ then for every $c$ you have that $a+c \rightarrow b+c$ and $c+a \rightarrow c+b$. There is no requirement for the converse implication to be true. 
Without the existence of a nullary operation you cannot necessarily obtain the other implication. 
In general, quasi ordered algebras only need each component of their operations to be either order preserving or order reversing, they don't need to be surjective on the order structure. 
