Galois connections between quasi orders?

Question

A binary relation over a set $A$ is a quasi order if it is reflexive and transitive. It would be a partial order if it also were anti-symmetric.

I've always seen Galois connections be defined over partially ordered sets. I can immagine a similar concept defined over quasi ordered sets as well.

Is the concept of Galois connection between quasi ordered sets meaningful/interesting? Has it been studied and if so where can I find some references?

Answer 1

Galois Connections Presented Calculationally by C.J. Aarts gives a wonderfully accessibly introduction to the theory along with clear proofs and flurry of examples.

It can be found online at: http://www.cs.nott.ac.uk/~psarb2/MPC/galois.ps.gz

Roland Backhouse has a host of material on Galois Connections and is promoter of their use in computing science. See his site for many lovely articles: http://www.cs.nott.ac.uk/~psarb2/papers/papers.html

Answer 2

I should think one of the most interesting references you can get about this is the paper

Urquhart, Alasdair. A topological representation theory for lattices. Algebra Universalis, 8, 45-58 (1978).

and maybe some papers that cite it.

There, the author uses, not exactly quasi-ordered sets, but rather doubly-ordered sets, which are structures comprised of two quasi-orderings, say $\leq_1$ and $\leq_2$, satisfying $$x \leq_1 y \quad\text{and}\quad x \leq_2 y \quad\text{implies}\quad x = y.$$

So suppose you have a doubly-ordered set $\langle X, \leq_1, \leq_2\rangle$.
The Galois connection is then given by the pair $\langle l,r \rangle$, where $$l(A) = \{ x \in X : \forall y \in X \,( x \leq_1 y \Rightarrow y \notin A ) \}$$ and $$r(A) = \{ x \in X : \forall y \in X \,( x \leq_2 y \Rightarrow y \notin A ) \}.$$

He uses these structures, equipped with a suited topology, to represent arbitrary bounded lattices, in a similar way as Hilary Priestley used posets to represent bounded distributive lattices.

This representation is not as well behaved as the one of distributive lattices.
In particular, it doesn't extend to a categorical duality (although it can represent onto homomorphisms, but even these in a way that is far from easy to follow, and perhaps much less useful).

You may find similarities to this theory in Formal Concept Analysis (without the topology).
There are yet other references to similar structures, but I can't find them now.