Longer chains of adjoints :-)

Question

The first question:

Is in true that there exist chains of Galois connections (let's limit to Galois connections between posets) of arbitrary lengths $n$?

$F_0(a) \leq b$ if and only if $a \leq F_1(b)$; ...; $F_{n-1}(a) \leq b$ if and only if $a \leq F_n(b)$.

Then (as it is well known) $F_i(a) = \min \{ c \mid a \leq F_{i+1}(c) \}$.

Second question:

What about (which properties it has?) continuing this chain for $i<0$ if we instead require the weaker condition: $F_i(a) = \inf \{ c \mid a \leq F_{i+1}(c) \}$?

And finally: What's about generalizing this to arbitrary adjoint functors?

Answer 1

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 ;)

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Perhaps you wanted an example in the non-discrete order?

No probs: take Fₙ to be the identity function ;)

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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!