Journals Information
Mathematics and Statistics Vol. 13(3), pp. 145 - 153
DOI: 10.13189/ms.2025.130303
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Cubic 
-subalgebras and Its Properties
Haliz J. Othman 1, Alias B. Khalaf 2,*
1 Basic Education College, Math Dept., Duhok University, Iraq
2 Math. Dept, Faculty of Science, Duhok University, Iraq
ABSTRACT
This article proposes the notion of cubic 
-subalgebras with the aid of cubic sets known as 
-cubic -subalgebras where 
. This paper investigates the fundamental properties and relationships between cubic sets and cubic -subalgebras, offering a classification framework and analyzing their significance within the broader context of algebraic systems. Additionally, we provide examples and describe the structure of cubic -subalgebras. We specify the conditions that are both necessary and sufficient for a cubic -subalgebra in order the crisp subsets to be -subalgebra. We proved that the 
-union of any collection of 
-cubic -subalgebras is a -cubic -subalgebra, while the intersection of type 
and 
-union of -cubic -subalgebras may not be an -cubic -subalgebra. Also, we proved that the subset 
is a -subalgebra under certain conditions. Moreover, some examples in 
with the usual subtraction are given to clarify the notion of the lower 
-level, the lower 
-level of a cubic sets and we showed that they are -subalgebras whenever their corresponding cubic set is a an -cubic -subalgebra. Finally, we proved that a cubic set 
is commutative cubic -subalgebra of 
if and only if it is -cubic -subalgebra when 
is a commutative -algebra.
KEYWORDS
Cubic Set, -algebra, Cubic -algebra
Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Haliz J. Othman , Alias B. Khalaf , "Cubic -subalgebras and Its Properties," Mathematics and Statistics, Vol. 13, No. 3, pp. 145 - 153, 2025. DOI: 10.13189/ms.2025.130303.
(b). APA Format:
Haliz J. Othman , Alias B. Khalaf (2025). Cubic -subalgebras and Its Properties. Mathematics and Statistics, 13(3), 145 - 153. DOI: 10.13189/ms.2025.130303.