Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-682213120220301On studying bi-\(\Gamma\)-algebra and some related concepts15651572577210.22075/ijnaa.2022.5772ENMohaimen M.AbboodMinistry of Education, General Directorate of Education in Diyala, IraqAli Al-Fayadh Ali Al-FayadhDepartment of Mathematics and Computer Applications, College of Science, Al-Nahraini University, Baghdad, IraqHassan H.EbrahimDepartment of Mathematics, College of Computer Science and Mathematics, Tikrit University, Tikrit, Salahaddin, IraqJournal Article20210904The aim of this research paper is to introduce the concept of bi-\(\Gamma\)-algebra space (bi-gamma algebra space). The concept of bi-\(\mu\)-measurable set in a bi-\(\Gamma\)-algebra space is defined. With this concept, some properties of bi-\(\Gamma\)-algebra space are proved. We then define various separation axioms for bi-\(\Gamma\)-algebra space such as \(M_{0},M_{1},\ M_{2},\ M_{3},\) and \(M_{4}\); then the relationships between them are studied. In addition, the concept of measurable function between two bi-measurable spaces is introduced and some results are discussed.https://ijnaa.semnan.ac.ir/article_5772_cb31ac7125b59a219f98eccaffbe1dd0.pdf