This sent me down a rabbit hole; thanks! ;)
You last two questions seem unclear to me.
However, for the first question: YES
For example, suppose f, g are inverse functions, ie
f x = y ≡ x = g y. Then this statement means that they
are Galois Connected in the discrete order, ie equality =.
Let Fₙ be f if n = 0 mod 2 and g otherwise.
Then, Fₙ x ≤ y ≡ x ≤ Fₙ₊₁ y and this is a chain ;)
Perhaps you wanted an example in the non-discrete order?
No probs: take Fₙ to be the identity function ;)
Perhaps you wanted an example in the non-discrete order
and with the non-identity function?
No probs: pick your favourite isotonic
(x ≤ y ≡ f x ≤ f y) and involutive (f ∘ f = id) function f,
then define Fₙ to be f.
Indeed, we have Fₙ x ≤ y ≡ x ≤ Fₙ₊₁ y as follows:
Fₙ x ≤ y
≡ “definition of Fₙ”
f x ≤ y
≡ “isotonicity of f”
f (f x) ≤ f y
≡ “f involoutive”
x ≤ f y
≡ “definition of Fₙ₊₁”
x ≤ Fₙ₊₁ y
Usually involutions tend to swap the order, and in some contexts
this is in-fact a definition of Galois Connection!! Weird..
( “A Glimpse Into The Wonderland Of Involutions”,
http://eqworld.ipmnet.ru/en/education/wiener.pdf )
Anyhow, examples include
• boolean negation: ¬ p ⇒ q ≡ p ⇐ ¬ q
• additive inverse: - x ≤ y ≡ x ≥ - y
• multiplicative inverse: 1 / x ≤ y ≡ x ≥ 1 / y
• the previous two: 1 / x ⊑ y ≡ x ⊑ 1 / y
where a ⊑ b ∶≡ -a ≤ b
• The above examples give a more general approach that avoids
the flipping of the order:
if f is an involution, then so is its “conjugate” F x ≔ - f ( - x)
---note that this applies to the three above, just use the appropriate negation
operator ;)
Moreover, if f is isotonic then so is F! Neato ;)
• F x ≔ a + (x - a)⁻¹ for any a, of course for domain we need x ≠ a;
like the above, this is order-reversing GC: F x ≤ y ≡ x ≥ F y.
(src: https://math.stackexchange.com/a/46814/80406 )
In-fact, this construction can be applied repetitively to yield new involutions:
if f is an involution, then so is F x ≔ f(x - a) + a;
moreover, if f is isotonic then so is F! Neato ;)
The above general construction are really instances of Babbage's Construction:
if f is an involution then so is
F ≔ φ⁻¹ ∘ f ∘ φ for any invertible function Φ.
(Src: http://www.jstor.org/stable/2007270?origin=crossref&seq=1#page_scan_tab_contents )
Moreover, if f and φ are isotonic then so is F!
Finally, before I leave you, consider this the more common or familiar is exponentiation and logarithms
functions. Both are monotonic and are inverses, whence they are isotonic.
( f x ≔ aˣ is isotonic and so is g x ≔ logₐ x, for a > 1. )
Then F x ≔ logₐ( (aˣ + d) / (aˣ - d) ) is isotonic if d < 0 otherwise flips-order (or so I claim!).
Moreover, F is probably involution; I haven't checked.
Anyhow, here's the challenge: is this scenario an instance of Babbage's Construction? Or, can you generalise this to arbitrary monotone bijection pairs? ;)
Good night!
