About definition of "ordered semi-ring"

Question

I need the definition of "ordered semi-ring". Can I use these properties:

$a \preceq b \to a + c \preceq b + c$

$0 \preceq a \wedge 0 ≤ b \to 0 \preceq a \cdot b$ (or: $a \preceq b \wedge 0 \preceq c \to a \cdot c \preceq b \cdot c $)

???

Thanks in advance!

Answer 1

I found three slightly different definitions of ordered semiring. This one may be the oldest:

Definition: A semiring $R$ is ordered if there is a partial order $\preceq$ on $R$ such that

1. $a+c\preceq b+c$ whenever $a,b,c\in R$ and $a\preceq b$, and
2. $ac\preceq bc$ and $ca\preceq cb$ whenever $a,b,c\in R$, $0\preceq c$, and $a\preceq b$.

$R$ is positive if $0\preceq a$ for all $a\in R$.

Another definition removes the restriction that $0\preceq c$ from (2); the two are clearly equivalent for positive ordered semirings. The third adds to the second the requirement that $0\preceq a$ for all $a\in R$, so it’s equivalent to the notion of positive ordered semiring in either of the first two senses of ordered semiring.

All of these definitions explicitly include additive monotonicity, your first property. Your second property follows from (2) above, so it holds no matter which of these definitions is in use: if $0\preceq a$ and $0\preceq b$, then $0=0\cdot b\preceq a\cdot b$. Thus, under any of these definitions of ordered ring you can use both of the properties that you listed.