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Minimal cohesive basic sets

Published online by Cambridge University Press:  20 January 2009

Donald L. Goldsmith
Donald L. Goldsmith
Affiliation:
Western Michigan University, Kalamazoo, Michigan 49001
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A basic set (formerly basic sequence) ℬ is a set of pairs (a, b) of positive integers satisfying

(1) if (a, b) ∈ ℬ, then (b, a) ∈ ℬ,

(2) (a, bc) ∈ ℬ if and only if (a, b) ∈ ℬ and (a, c) ∈ ℬ,

(3) (1, k ∈ ℬ, k = 1, 2, ….

Some familiar examples of basic sets are

, where Sk = {(1, k), (k, 1)},

= {(a, b)| a and b are relatively prime positive integers},

ℒ = {(a, b)| a and b are any positive integers}.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 1 , March 1974 , pp. 73 - 75
Copyright
Copyright © Edinburgh Mathematical Society 1974

References

REFERENCES

(1) Goldsmith, D. L., On the multiplicative properties of arithmetic functions, Pacific J. Math. 27 (1968), 283–304.CrossRefGoogle Scholar
(2) Goldsmith, D. L., On the structure of certain basic sequences associated with an arithmetic function, Proc. Edinburgh Math. Soc. 17 (1971), 305–310.CrossRefGoogle Scholar
(3) Gioia, A. A. and Goldsmith, D. L., Convolutions of arithmetic functions over cohesive basic sequences, Pacific J. Math. 38 (1971), 391–399.CrossRefGoogle Scholar