Normal Bipolar Soft Subgroups

Abstract

In this study, we introduce the concept of normal bipolar soft subgroup and investigate some properties of normal bipolar soft subgroups. We also define notions of bipolar soft left and right cosets of a bipolar soft group and the concept of conjugate bipolar soft subgroup, and obtain some of their properties.

Keywords:

• Bipolar soft group
• bipolar soft level set
• normal bipolar soft subgroup
• coset
• conjugate bipolar soft subgroup

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Faruk Karaaslan

Dr. Faruk Karaaslan graduated from the Department of Mathematics, Erciyes University, Kayseri, Turkey, in 2003 (BSc). He earned his MSc and PhD degrees from Gaziosmanpaşa University, Tokat, Turkey, in 2007 and 2013. Dr Karaaslan is currently an Associate Professor at Department of Mathematics, Faculty of Science, Çankırı Karatekin University. He has contributed more than 55 research papers in reputable journals and 4 book section in reputable publishers. His research interests include mathematical theories for modelling uncertainty, fuzzy sets, soft sets, rough sets, intuitionistic fuzzy sets, neutrosophic sets and bipolar fuzzy sets. He also studies on decision making, algebraic structures, and graphs structures of the mentioned set theories.

Aman Ullah

Dr. Aman Ullah graduated from the Department of Mathematics, University of Peshawar, Pakistan in 2001. He earned his MPhil and PhD degrees from University of Malakand, Pakistan in 2008 and 2016. Currently, Dr. Aman Ullah is serving as an Assistant Professor at Department of Mathematics, University of Malakand, Pakistan. He has contributed more than 40 research papers in reputable journals and 1 book section with a reputable publisher. His research interests include fuzzy algebra, noncommutative algebra, soft set theory, fractional differential equations, mathematical modelling and fuzzy fractional and integers order differential equation.

Imtiaz Ahmad

Dr. Imtiaz Ahmad graduated from Pakistani institutions. He earned PhD degrees from the Department of Mathematical Sciences, University of Essex, United Kingdom in 2010. Currently, Dr. Ahmad is serving as an Associate Professor at Department of Mathematics, University of Malakand, Pakistan, previously he served this department as Lecturer and Assistant Professor since February 2003. He published more than 50 research articles in various national and international research journals, writing general articles for newspapers, supervised several MPhil and PhDs, have completed different research projects, and authored several textbooks.