Give an example of a nonconstant function f from poset A to poset B which is simultaneously increasing and decreasing.

Question

I'm also asked to show that no such function exists if A is totally ordered.

I have no clue how to do this; how can a function even be simultaneously increasing and decreasing?

Thank you!

Answer 1

Assume a <= b in A. Then for such a function f,
f(a) <= f(b) <= f(a) and equality ensues.

This shows for linearily ordered A, f is constant.

Now let A = B = {a, b, c, d} be ordered by a < b, s < t.
Map s,t to t and a,b to b for a nonconstant function that
is both increasing and decreasing.

It's a got ya acoming an' a going function.
Clearly increasing means ascending (not strictly increasing).