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