[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\$$