[Matthew Doty #10](https://forum.azimuthproject.org/discussion/comment/16506/#Comment_16506) Thanks, very enlightening, especially using different notation for the two orders instead of e.g. \\(\sqsubseteq_A, \sqsubseteq_B\\). Typo: the orders need to be swapped in the definitions 1 - 4.

> I actually see 4 equivalent definitions of a Galois connection \\(f \dashv g\\) for two preorders \\(\langle A, \sqsubseteq\rangle\\) and \\(\langle B, \preceq\rangle\\)

So:

(1) \\(f(a) \preceq b\\) if and only if \\(a \sqsubseteq g(b)\\)

(2) \\(f\\) and \\(g\\) are mono and \\(f(g(b)) \preceq b\\) and \\(a \sqsubseteq g(f(a))\\)

(3) \\(f\\) is mono and \\(f(g(b)) \preceq b\\) and \\(f(a) \preceq b \Longrightarrow a \sqsubseteq g(b)\\)

(4) \\(g\\) is mono and \\(a \sqsubseteq g(f(a))\\) and \\(a \sqsubseteq g(b) \Longrightarrow f(a) \preceq b\\)