Power GCD and power LCM matrices defined on GCD-closed sets over unique factorization domains

Y. A. Awad, T. Kadri and R. H. Mghames
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 1, Pages 150–166
DOI: 10.7546/nntdm.2019.25.1.150-166
Full paper (PDF, 225 Kb)

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Authors and affiliations

Y. A. Awad
School of Arts and Sciences, Department of Mathematics and Physics
Lebanese International University, Bekaa, Lebanon

T. Kadri
School of Arts and Sciences, Department of Mathematics and Physics
Lebanese International University, Bekaa, Lebanon

R. H. Mghames
School of Arts and Sciences, Department of Mathematics and Physics
Lebanese International University, Bekaa, Lebanon

Abstract

Let T = {t₁, t₂, …, t_m} be a well ordered set of m distinct positive integers with t₁ < t₂ < … < t_m. The GCD matrix on T is defined as (T)_m×m = (t_i, t_j), where (t_i, t_j) is the greatest common divisor of t_i and t_j , and the power GCD matrix on T is (T^r)_m×m = (t_i, t_j)^r, where r is any real number. The LCM matrix on T is defined as [T]_m×m = [t_i, t_j], where [t_i, t_j] is the least common multiple of t_i and t_j, and the power LCM matrix on T is [T^r]_m×m = [t_i, t_j]^r. Set T = {t₁, t₂, …, t_m} is said to be gcd-closed if (t_i, t_j) ∈ T for every t_i and t_j in T. In this paper, we give a generalization for the power GCD and LCM matrices defined on gcd-closed sets over unique factorization domains (UFDs). Moreover, we present a speculation for a generalization of Bourque–Ligh conjecture to UFDs which states that the least common multiple matrix defined on a gcd-closed P-ordered set in any UFD is nonsingular. Some examples that show what is done are additionally given in ℤ[i] and ℤ_p[x].

Keywords

• Power GCD P-matrix
• Power LCM P-Matrix
• P-ordering
• gcd-closed sets
• Prime residue system
• Unique factorization domains

2010 Mathematics Subject Classification

• Primary
  • 11C20
  • 11A25
• Secondary
  • 13F15
  • 15A36
  • 16U30

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