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Niceness theorems

In: Advances in Combinatorial Mathematics

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  • Michiel Hazewinkel

Abstract

There are many results and constructions in mathematics that are * unreasonably nice *. For instance it appears to be difficult for a set to carry many compatible (algebraic) structures. More precisely, if, say, an algebra carries a compatible *higher* structure the underlying algebra must be very regular. For instance, if an associative unital algebra (over a characteristic zero field) carries a graded connected Hopf algebra structure the underlying algebra is free commutative. There are many such theorems in various different parts of mathematics. This paper gives a number of examples of this phenomenon and of similar phenomena as a preliminary step in starting to examine and try to understand this matter. Besides unreasonably nice constructions and theorems there is also the matter of nice proofs. By this I mainly mean proofs that principally rely on, for instance, the universal properties that define an object, and that do not rely (too much) on calculations. This matter is touched upon in the last section of this paper.

Suggested Citation

  • Michiel Hazewinkel, 2009. "Niceness theorems," Springer Books, in: Ilias S. Kotsireas & Eugene V. Zima (ed.), Advances in Combinatorial Mathematics, chapter 0, pages 79-125, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-03562-3_5
    DOI: 10.1007/978-3-642-03562-3_5
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