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

### References

- Beslin, S., & El-Kassar, A. N. (1989). GCD matrices and Smith’s determinant over U.F.D., Bull. Number Theory Related Topics, 13, 17–22.
- Beslin, S., & Ligh, S. (1989). Greatest common divisor matrices, Linear Algebra and Its Applications, 118, 69–76.
- Beslin, S. (1991). Reciprocal GCD matrices and LCM matrices, Fibonacci Quart., 20, 71–274.
- Beslin, S., & Ligh, S. (1989). Another generalization of Smith’s determinant, Bull Australian Math. Soc., 40, 413–415.
- Beslin, S., & Ligh, S. (1992). GCD-closed sets and the determinants of GCD matrices, Fibonacci Quart., 30, 157–160.
- Borque, K., & Ligh, S. (1992). On GCD and LCM matrices, Linear Algebra and Its Applications, 174, 65–74.
- Chun, S. Z. (1996). GCD and LCM Power Matrices, Fibonacci Quart., 43, 290–297.
- El-Kassar, A. N., Awad, Y. A., & Habre, S. S. (2009). GCD and LCM matrices on factor closed sets defined in principle ideal domains, Journ. Math. and Stat., 5, 342–347.
- El-Kassar, A. N., Habre, S. S., & Awad, Y. A. (2010). GCD Matrices Defined on gcd-closed Sets in a PID, International Journal of Applied Mathematics, 23, 571–581.
- Haukkanen, P. (1997). On Smith’s determinant, Linear Algebra and Its Applications, 258, 251–269.
- Haukkanen, P., & Sillanpää, J. (1997). On some analogues of the Bourque–Ligh conjecture on LCM matrices, Notes on Number Theory and Discrete Mathematics, 3 (1), 52–57.
- Hong, S. (1998). On LCM matrices on GCD-closed sets (English summary), Southeast Asian Bull. Math., 22, 381–384.
- Hong, S. (1998). Bounds for determinant of matrices associated with classes of arithmetical functions, Linear Algebra and Its Applications, 281, 311–322.
- Hong, S. (2004). Asymptotic behavior of eigen values of GCD power matrices, Glasgow Math. J., 46, 551–569.
- Hong, S., Zhou, X., & Zhao, J. (2009). Power GCD Matrices for a UFD, Algebra Colloquium, 16, 71–78.
- Li, Z. (1990). The determinant of a GCD matrices. Lin. and Multilin. Alg., 134, 137–143.
- Ligh, S. (1988). Generalized Smith’s determinant, Lin. and Multilin. Alg., 22, 305–306.
- Smith, H. J. S. (1875/76). On the value of a certain arithmetical determinant, Proc. London Math. Soc., 7, 208–212.

## Related papers

## Cite this paper

APAAwad, 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.

ChicagoAwad, Y. A., T. Kadri and R. H. Mghames. “Power GCD and Power LCM Matrices Defined on GCD-closed Sets Over Unique Factorization Domains.” Notes on Number Theory and Discrete Mathematics 25, no. 1 (2019): 150-166, doi: 10.7546/nntdm.2019.25.1.150-166.

MLAAwad, Y. A., T. Kadri and R. H. Mghames. “Power GCD and Power LCM Matrices Defined on GCD-closed Sets Over Unique Factorization Domains.” Notes on Number Theory and Discrete Mathematics 25.1 (2019): 150-166. Print, doi: 10.7546/nntdm.2019.25.1.150-166.