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"... this paper is as follows. In the second section I present the Aristotle-Wille theory of abstract concepts, including a proof of Wille's theorem constructing the complete lattice of concepts for an arbitary relation. In the third section I apply this theory to symmetric relations; in the fourth ..."

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this paper is as follows. In the second section I present the Aristotle-Wille theory of abstract concepts, including a proof of Wille's theorem constructing the complete lattice of concepts for an arbitary relation. In the third section I apply this theory to symmetric relations; in the fourth section I discuss transitive relations; in the fth section I discuss more general pairings. The examples are drawn from analysis, operator theory, algebra, dynamical systems, and even elementary number theory. They include some of the most important examples of \duality" in mathematics; a few are merely personal favorites of the author. For each example, I rst give the relation which gives rise to a concept lattice (or, equivalently, a pair of lattices in galois duality), and then briey discuss some of the more important features of the concept lattice