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

Eplett, W. J. R.

doi:10.1090/S0002-9939-1981-0589130-X

An additive representation for real functions on the product of a set and a lattice
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by W. J. R. Eplett PDF

Proc. Amer. Math. Soc. 81 (1981), 23-26 Request permission

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

Peter C. Fishburn, Additive representations of real functions on product sets, J. Combinatorial Theory 4 (1968), 397–402. MR 218251
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 174487, DOI 10.1007/BF00531932