Sains Malaysiana 48(12)(2019): 2817–2830

http://dx.doi.org/10.17576/jsm-2019-4812-23

A New Classification of Hemirings through Double-Framed Soft h-Ideals

(Pengelasan Baru Hemirings melalui h-Ideals Lembut-Dua Kerangka)

FAIZ MUHAMMAD KHAN^1,4*,NIEYUFENGU¹, HIDAYAT ULLAH KHAN² & ASGHAR KHAN³

¹Department of Applied Mathematics, School of Natural and Applied Sciences, Northwestern Polytechnical University Xi'an, Shaanxi, PR China

²Department of Mathematics, University of Malakand, Lower Dir Chakddara, KP, Pakistan

³Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, KP, Pakistan

⁴Department of Mathematics and Statistics, University of Swat, KP, Pakistan

Diserahkan: 21 Februari 2019/Diterima: 23 Disember 2019

ABSTRACT

Due to lack of parameterization, various ordinary uncertainty theories like theory of fuzzy sets, and theory of probability cannot solve complicated problems of economics and engineering involving uncertainties. The aim of the present paper was to provide an appropriate mathematical tool for solving such type of complicated problems. For the said purpose, the notion of double-framed soft sets in hemirings is introduced. As h-ideals of hemirings play a central role in the structural theory, therefore, we developed a new type of subsystem of hemirings. Double-framed soft left (right) h-ideals, double-framed soft h-bi-ideals and double-framed soft h-quasi-ideals of hemiring R are determined. These concepts are elaborated through suitable examples. Furthermore, we are bridging ordinary h-ideals and double-framed soft h-ideals of hemirings through double-framed soft including sets and characteristic double-framed soft functions. It is also shown that every double-framed soft h-quasi-ideal is double-framed soft h-bi-ideal but the converse inclusion does not hold. A well-known class of hemrings i.e. h-hemiregular hemirings is characterized by the properties of these newly developed double-framed soft h-ideals of R.

Keywords: DFS h-bi-ideal; DFS h-hemiregularhemirin; DFS h-quasi-idealg; DFS sets; h-ideals

ABSTRAK

Disebabkan oleh kekurangan pemparameteran, pelbagai teori ketidakpastian biasa seperti teori set kabur dan teori kebarangkalian tidak boleh menyelesaikan masalah ekonomi dan kejuruteraan yang rumit yang melibatkan ketidakpastian. Tujuan penulisan kertas ini adalah untuk menyediakan satu alat matematik yang sesuai untuk menyelesaikan masalah rumit yang sedemikian. Untuk tujuan tersebut, satu tanggapan set lembut dual kerangka dalam hemirings diperkenalkan. Oleh kerana h-ideals hemiring memainkan peranan utama dalam teori struktur, maka kami telah membangunkan satu jenis subsistem hemiring baru. h-ideals lembut kiri (kanan) dual kerangka, h-dwi-ideal lembut dual kerangka dan h-separa-ideal lembut dual kerangka hemirings ditentukan. Konsep ini dihuraikan melalui contoh yang sesuai. Selain itu, kami menghubungkan h-ideals biasa dan h-ideals lembut dual kerangka hemirings melalui set lembut dual kerangka dan pencirian fungsi lembut dual kerangka. Kajian ini juga menunjukkan bahawa setiap h-quasi-ideal lembut dual bingkai adalah h-dwi-ideal lembut dual kerangka tetapi rangkuman akas tidak dapat bertahan. Satu kelas hemirings terkenal iaitu h-hemisekata hemirings dicirikan oleh sifat h-ideals dua bingkai lembut daripada R yang baru dibangunkan ini.

Kata kunci: Set DFS; DFS h-dwi-ideal; DFS h-hemisekata hemiring; h-ideal; DFS h-separa-ideal

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*Pengarang untuk surat-menyurat; email: faiz_zady@yahoo.com