Exponentiation and Euler measure
 James Propp
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Two of the pillars of combinatorics are the notion of choosing an arbitrary subset of a set with n elements (which can be done in 2^{ n } ways), and the notion of choosing a kelement subset of a set with n elements (which can be done in \( n \choose k \) ways). In this article I sketch the beginnings of a theory that would import these notions into the category of hedral sets in the sense of Morelli and the category of polyhedral sets in the sense of Schanuel. Both of these theories can be viewed as extensions of the theory of finite sets and mappings between finite sets, with the concept of cardinality being replaced by the more general notion of Euler measure (sometimes called combinatorial Euler characteristic). I prove a “functoriality” theorem (Theorem 1) for subsetselection in the context of polyhedral sets, which provides quasicombinatorial interpretations of assertions such as \( 2^{1} = {1 \over 2} \quad\mathrm{and}\quad{{1/2} \choose {2}} = {1 \over 8} \) . Furthermore, the operation of forming a power set can be viewed as a special case of the operation of forming the set of all mappings from one set to another; I conclude the article with a polyhedral analogue of the set of all mappings between two finite sets, and a restrictive but suggestive result (Theorem 2) that offers a hint of what a general theory of exponentiation in the polyhedral category might look like. (Other glimpses into the theory may be found in [11].)
 Title
 Exponentiation and Euler measure
 Journal

algebra universalis
Volume 49, Issue 4 , pp 459471
 Cover Date
 20031001
 DOI
 10.1007/s0001200318171
 Print ISSN
 00025240
 Online ISSN
 14208911
 Publisher
 BirkhäuserVerlag
 Additional Links
 Keywords

 05A15, 52B05.
 Keywords:
 Euler characteristic
 polyhedral sets
 Authors

 James Propp ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Wisconsin, ., 53706, Madison, Wisconsin, USA