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Minimal cohesive basic sets
Published online by Cambridge University Press: 20 January 2009
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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}.
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- Research Article
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- 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
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