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An additive representation for real functions on the product of a set and a lattice

Author: W. J. R. Eplett
Journal: Proc. Amer. Math. Soc. 81 (1981), 23-26
MSC: Primary 06A15
MathSciNet review: 589130
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Abstract: Given a real-valued function defined on the product of an arbitrary set and a finite lattice, a necessary and sufficient condition is obtained for the existence of an additive representation for the function in terms of functions on sublattices of the original lattice. This additive representation is of the nature of a recurrence and provides a tool for further analysis of the function. An example is given for a case where the lattice concerned is the lattice of partitions of a finite set. The main theorem of this paper generalizes a result due to Fishburn corresponding to the lattice being the lattice of subsets of a finite set.

References [Enhancements On Off] (What's this?)

  • [1] Peter C. Fishburn, Additive representations of real functions on product sets, J. Combinatorial Theory 4 (1968), 397–402. MR 0218251
  • [2] Gian-Carlo Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368 (1964). MR 0174487

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Article copyright: © Copyright 1981 American Mathematical Society