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

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## Details

### 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* = {*t*_{1}, *t*_{2}, …, *t _{m}*} be a well ordered set of

*m*distinct positive integers with

*t*

_{1}<

*t*

_{2}< … <

*t*. The GCD matrix on

_{m}*T*is defined as (

*T*)

_{m×m}= (

*t*), where (

_{i}, t_{j}*t*) is the greatest common divisor of

_{i}, t_{j}*t*and

_{i}*t*, and the power GCD matrix on

_{j}*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*], where [

_{i}, t_{j}*t*] is the least common multiple of

_{i}, t_{j}*t*and

_{i}*t*, and the power LCM matrix on

_{j}*T*is [

*T*]

^{r}_{m×m}= [

*t*]

_{i}, t_{j}^{r}. Set

*T*= {

*t*

_{1},

*t*

_{2}, …,

*t*} is said to be gcd-closed if (

_{m}*t*) ∈

_{i}, t_{j}*T*for every

*t*in

_{i}and t_{j}*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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## Related papers

## Cite this paper

Awad, Y. A., Kadri, T., & Mghames, R. H. (2019). Power GCD and power LCM matrices defined on GCD-closed sets over unique factorization domains. Notes on Number Theory and Discrete Mathematics, 25(1), 150-166, doi: 10.7546/nntdm.2019.25.1.150-166.