I am working in a space $V$ of objects that behaves like a vector space with a partial ordering $\preceq$. I have discovered an operator $f:V\times V \rightarrow V$ with the following properties:
For any $a,b,c \in V$ and any scalar $n$:
- $f(a,a) = a$
- $f(a,b) = f(b,a)$
- $f(a+c, b+c) = f(a,b) + c$
- $f(n\cdot a,n\cdot b) = n\cdot f(a,b)$.
- $a \preceq f(a,b)$.
- If $a \preceq b$, then $f(a,b) = b$.
- If $a\preceq c$ and $b \preceq c$ then $f(a,b) \preceq c$.
I have two questions. First, besides the first two properties (idempotence, symmetry), I'm not sure whether these properties have official names.
Second, I'm wondering whether operators with these properties are familiar in other contexts.