Let $X,Y$ be two posets, and let $f:X\to Y$ and $g:Y\to X$ be two maps giving rise to an antitone Galois connection. Assuming that $X,Y$ are lattices, how do I prove that meets and joins are reversed by $f$? (I'm stuck with one of four inequalities).
$f(x\wedge x')=f(x)\vee f(x')$. I could prove one direction: $x\wedge x'\leq x,x'$, so $f(x\wedge x')\geq f(x),f(x')$, and by the universal property $f(x\wedge x')\geq f(x)\vee f(x')$. I don't get how to prove the converse though; would you only give me a hint?
$f(x\vee x')=f(x)\wedge f(x')$. Similarly to above $f(x\vee x')\leq f(x)\wedge f(x')$. Conversely $f(x\vee x')\geq f(x)\wedge f(x')$ $\iff$ $x\vee x'\leq g(f(x)\wedge f(x'))$. By symmetry $g(f(x)\wedge f(x'))\geq gf(x)\vee gf(x')$, so it is sufficient that $x\vee x'\leq gf(x)\vee gf(x')$, which is easy to show.
