@MichaelGreinecker Generalized isotonic mappings of sets are functions that preserve the order of sets, but they also allow the possibility of incomparability. More formally, a function f: X → Y is isotonic if for any subsets A, B of X, if A ⊆ B and there is no x in A and y in B such that x is not comparable to y, then f(A) ⊆ f(B).

@YCor I used a more general term due to my own inexperience. studying mathematics in my undergraduate degree, I did not come across such terms even in my native language, and my technical English is not good enough for a correct translation. Sorry about that, "isotonic maps between posets" is indeed more correct

@YCor An isotonic mapping is a function between two partially ordered sets that preserves the ordering between the elements. Specifically, given two partially ordered sets (X, ≤) and (Y, ≤), a function f: X → Y is isotonic if for any x, y ∈ X such that x ≤ y, it is true that f(x) ≤ f(y).