Is there a standard name for a function that maps the minimum of the domain to the minimum of the codomain?
In greater detail, let $\mathbf{X} = (X,\leq)$ and $\mathbf{Y} = (Y, \leq)$ be partially-ordered spaces. Suppose $\mathbf{X}$ and $\mathbf{Y}$ each has a minimum. Let $f:X\rightarrow Y$. Is there a name, known in the mathematical literature, for the following property of $f$: $ f(\min \mathbf{X}) = \min\mathbf{Y}$?
I would probably use the phrase "minimum-preserving" for such functions, but would define what that means in context. The phrase is used in other mathematical texts*, although I couldn't find one using precisely this definition.
*See the book "Cooperative Control of Dynamical Systems: Applications to Autonomous Vehicles" by Zhihua Qu, the blog post "The Smooth-Max Minimum Incident of December 2018" by Erik Erlandson, and the MathOverflow post Minimum Preserving Transformations.