2
$\begingroup$

An isotonic mapping is a function between two partially ordered sets that preserves the ordering between the elements. Specifically, given two partially ordered sets $(X,\le)$ and $(Y,\le)$, a function $f:X\to Y$ is isotonic if for any $x, y\in X$ such that $x\le y$, it is true that $f(x) \le f(y)$.

I am looking for literature recommendations on classical and generalized isotonic mappings of sets. Specifically, I am interested in any scientific and technical literature on this topic that is available in English or Russian (other languages are also acceptable). I am open to both books and articles.

If anyone is knowledgeable in this area, could you suggest where this theory is actively used? Specifically, I am interested in learning about the specific areas of mathematics and algorithms that this theory is used in.

Thank you in advance for any help you can provide.

Improve this question
$\endgroup$
10
  • 1
    $\begingroup$ Could you provide at least one link using this terminology? The links I've found are totally unrelated to set theory. $\endgroup$
    – YCor
    Commented Mar 18, 2023 at 14:37
  • $\begingroup$ This is the context in which I know isotonic, yours must be different? $\endgroup$
    – kjetil b halvorsen
    Commented Mar 18, 2023 at 14:45
  • 1
    $\begingroup$ @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). $\endgroup$
    – stalinon
    Commented Mar 19, 2023 at 10:21
  • 1
    $\begingroup$ That leaves open the question of what you mean by generalized isotonic mappings. $\endgroup$
    – Michael Greinecker
    Commented Mar 19, 2023 at 10:33
  • 2
    $\begingroup$ Every function $f:X\to Y$ satisfies that $A\subseteq B$ implies $f(A)\subseteq f(B)$ without any further condition. $\endgroup$
    – Michael Greinecker
    Commented Mar 19, 2023 at 10:55

1 Answer 1

Reset to default
1
$\begingroup$

The monotonic functions are the:

  • nondecreasing functions. They are given by the isotonic mappings $(X,\le)$ and $(Y, \le)$ since $f(x)\le f(y)$ if $x\le y$.

  • nonincreasing functions. They are given by the isotonic mappings $(X,\ge)$ and $(Y, \le)$ since $f(x)\le f(y)$ if $x\ge y$.

  • decreasing functions. They are given by the isotonic mappings $(X,\gt)$ and $(Y, \lt)$ since $f(x)\lt f(y)$ if $x\gt y$.

  • increasing functions. They are given by the isotonic mappings $(X,\lt)$ and $(Y, \lt)$ since $f(x)\lt f(y)$ if $x\lt y$.

The isotonic mappings are mainly used in the functional analysis to prove the monotonicity of some functions.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .