Conditions for a neutrosophic set to be a (q, ∈∨q)-neutrosophic subalgebra are considered.

Saeid and Jun introduced the notion of neutrosophic points, and studied neutrosophic subalgebras of several types in BCK=BCI-algebras by using the notion of neutrosophic points.

The concept of neutrosophic set (NS) developed by Smarandache is a more general platform which extends the concepts of the classic set and fuzzy set , intuitionistic fuzzy set and interval valued intuitionistic fuzzy set.

The notion of (i; j; k)-length neutrosophic subalgebras in BCK/BCI-algebras is introduced, and their properties are investigated. Characterizations of length neutrosophic subalgebras are discussed by using level sets of interval neutrosophic sets. Conditions for level sets of interval neutrosophic sets to be subalgebras are provided.

The notion of a neutrosophic quadruple BCK/BCI-number is considered, and a neutrosophic quadruple BCK/BCI-algebra, which consists of neutrosophic quadruple BCK/BCI-numbers, is constructed.

In the fuzzy set which is introduced by Zadeh, the membership degree is expressed by only one function so called the truth function. As a generalization of fuzzy set, intuitionistic fuzzy set is introduced by Atanassove by using membership function and nonmembership function.

In this paper, we propose the notion of single-valued neutrosophic soft topological K-algebras. We discuss certain concepts, including interior, closure, C5-connected, super connected, Compactness and Hausdorff in singlevalued neutrosophic soft topological K-algebras. We illustrate these concepts with examples and investigate some of their related properties. We also study image and pre-image of single-valued neutrosophic soft topologicalK-algebras.

Commutative neutrosophic quadruple ideals and BCK-algebras are discussed, and related properties are investigated. Conditions for the neutrosophic quadruple BCK-algebra to be commutative are considered. Given subsets A and B of a neutrosophic quadruple BCK-algebra, conditions for the set NQ(A;B) to be a commutative ideal of a neutrosophic quadruple BCK-algebra are provided.

In this paper, we introduce the notion of MBJneutrosophic sub-algebra and MBJ-neutrosophic lter on equality algebras and investigate some equivalence de finitions, properties and the relation between them. Also, by using the notion of MBJ-neutrosophic lter, we introduce a congruence relation on equality algebra and show that the quotient is an equality algebra.

This ninth volume of Collected Papers includes 87 papers comprising 982 pages on Neutrosophic Theory and its applications in Algebra, written between 2014-2022 by the author alone or in collaboration with the following 81 co-authors (alphabetically ordered) from 19 countries: E.O. Adeleke, A.A.A. Agboola, Ahmed B. Al-Nafee, Ahmed Mostafa Khalil, Akbar Rezaei, S.A. Akinleye, Ali Hassan, Mumtaz Ali, Rajab Ali Borzooei , Assia Bakali, Cenap Özel, Victor Christianto, Chunxin Bo, Rakhal Das, Bijan Davvaz, R. Dhavaseelan, B. Elavarasan, Fahad Alsharari, T. Gharibah, Hina Gulzar, Hashem Bordbar, Le Hoang Son, Emmanuel Ilojide, Tèmítópé Gbóláhàn Jaíyéolá, M. Karthika, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Huma Khan, Madad Khan, Mohsin Khan, Hee Sik Kim, Seon Jeong Kim, Valeri Kromov, R. M. Latif, Madeleine Al-Tahan, Mehmat Ali Ozturk, Minghao Hu, S. Mirvakili, Mohammad Abobala, Mohammad Hamidi, Mohammed Abdel-Sattar, Mohammed A. Al Shumrani, Mohamed Talea, Muhammad Akram, Muhammad Aslam, Muhammad Aslam Malik, Muhammad Gulistan, Muhammad Shabir, G. Muhiuddin, Memudu Olaposi Olatinwo, Osman Anis, Choonkil Park, M. Parimala, Ping Li, K. Porselvi, D. Preethi, S. Rajareega, N. Rajesh, Udhayakumar Ramalingam, Riad K. Al-Hamido, Yaser Saber, Arsham Borumand Saeid, Saeid Jafari, Said Broumi, A.A. Salama, Ganeshsree Selvachandran, Songtao Shao, Seok-Zun Song, Tahsin Oner, M. Mohseni Takallo, Binod Chandra Tripathy, Tugce Katican, J. Vimala, Xiaohong Zhang, Xiaoyan Mao, Xiaoying Wu, Xingliang Liang, Xin Zhou, Yingcang Ma, Young Bae Jun, Juanjuan Zhang.