### 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
$$ (\forall a,b,c \in S) \qquad a \to b \implies a + c \to b + c $$



Is this a pre-ordered commutative semigroup?
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.


