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Arithmetical Functions and Distributivity

Published online by Cambridge University Press:  20 November 2018

P. J. McCarthy
P. J. McCarthy*
Affiliation:
University of Kansas, Lawrence, Kansas
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In this note we shall present a result about incidence functions on a locally finite partially ordered set, a result which is related to theorems of Lambek [2] and Subbarao [6]. Our terminology and notation will be that of Smith [4, 5] and Rota [7].

Let (L, ≤) be a partially ordered set which is locally finite in the sense that for all x, y ∊ L the interval [x, y] = {z | x ≤ z ≤ y} is finite. Denote by A(L, ≤) the set of functions f from L × L into some field, which is fixed once and for all, such that f(x, y) = 0 whenever x ≰ y.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 4 , December 1970 , pp. 491 - 496
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Carlitz, L., Arithmetic functions in an unusual setting, Amer. Math. Monthly 73 (1966), 582-590.Google Scholar
2. Lambek, J., Arithmetical functions and distributivity, Amer. Math. Monthly 73 (1966), 969-973.Google Scholar
3. Narkiewicz, W., On a class of arithmetical convolutions, Colloq. Math. 10 (1963), 81-94.Google Scholar
4. Smith, David, Incidence functions as generalized arithmetic functions, I, Duke Math. J. 36 (1967), 617-634.Google Scholar
5. Smith, David Incidence functions as generalized arithmetic functions, III, Duke Math. J. 36 (1969), 353-368.Google Scholar
6. Subbarao, M. V., Arithmetic functions and distributivity, Amer. Math. Monthl. 75 (1968), 984-989.Google Scholar
7. Rota, Gian-Carlo, On the foundations of combinatorial theory, I, Theory of M?bius functions, Z. Wahrsch. 2 (1964), 340-368.Google Scholar
8. Szász, Gabor, Introduction to lattice theory, Academic Press, New York, 1963.Google Scholar