... we focus on the canonical extension of partially ordered sets, which was defined by algebraic means by Dunn, Gehrke and Palmigiano . We show that it can be obtained alternatively via a generalization of Urquhart and Hartung’s maximal filter-ideal pair construction (, ). We further give a first-order dual characterization of perfect lattice hemiand homomorphisms, in the spirit of, but different from Gehrke , and make category-theoretic observations regarding the canonical extension. The second part of the thesis concerns the algebraic canonicity proof of the Sahlqvist fragment for distributive modal logic by Gehrke, Nagahashi and Venema . We pay particular attention to the additional operation n, which is crucial to that proof, and show that the proof can not be straightforwardly translated to an algebraic canonicity proof of the inductive fragment for distributive modal logic . We extract requirements on a new version of the operation n, which would yield a proof of the canonicity of the inductive fragment, and finish by starting to explore two new perspectives on the magical nature of the operation n
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