The Implementation of Rough Set on A Group Structure

Let  be a non-empty set and  an equivalence relation on . Then,  is called an approximation space. The equivalence relation on  forms disjoint equivalence classes. If , then we can form a lower approximation and an upper approximation of . If X⊆U, then we can form a lower approximation and an upper approximation of X. In this research, rough group and rough subgroups are constructed in the approximation space  for commutative and non-commutative binary operations

Set-valued mapping and Rough Probability

In 1982, the theory of rough sets proposed by Pawlak and in 2013, Luay concerned a rough probability by using the notion of Topology. In this paper, we study the rough probability in the stochastic approximation spaces by using set-valued mapping and obtain results on rough expectation, and rough variance.Comment: 9 page

2-Absorbing Vague Weakly Complete Γ-Ideals in Γ-Rings

The aim of this study is to provide a generalization of prime vague Γ-ideals in Γ-rings by introducing non-symmetric 2-absorbing vague weakly complete Γ-ideals of commutative Γ-rings. A novel algebraic structure of a primary vague Γ-ideal of a commutative Γ-ring is presented by 2-absorbing weakly complete primary ideal theory. The approach of non-symmetric 2-absorbing K-vague Γ-ideals of Γ-rings are examined and the relation between a level subset of 2-absorbing vague weakly complete Γ-ideals and 2-absorbing Γ-ideals is given. The image and inverse image of a 2-absorbing vague weakly complete Γ-ideal of a Γ-ring and 2-absorbing K-vague Γ-ideal of a Γ-ring are studied and a 1-1 inclusion-preserving correspondence theorem is given. A vague quotient Γ-ring of R induced by a 2-absorbing vague weakly complete Γ-ideal of a 2-absorbing Γ-ring is characterized, and a diagram is obtained that shows the relationship between these concepts with a 2-absorbing Γ-ideal

History and new possible research directions of hyperstructures

We present a summary of the origins and current developments of the theory of algebraic hyperstructures. We also sketch some possible lines of research

Stone-type representations and dualities for varieties of bisemilattices

In this article we will focus our attention on the variety of distributive bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and involutive bisemilattices. After extending Balbes' representation theorem to bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn duality and introduce the categories of 2spaces and 2spaces$^{\star}$. The categories of 2spaces and 2spaces$^{\star}$ will play with respect to the categories of distributive bisemilattices and De Morgan bisemilattices, respectively, a role analogous to the category of Stone spaces with respect to the category of Boolean algebras. Actually, the aim of this work is to show that these categories are, in fact, dually equivalent

A more general framework than the delta-primary hyperideals

In this paper we aim to study the notion of (t,n)-absorbing delta-semiprimary hyperideal in a Krasner (m,n)-hyperring

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

Collected Papers (on Neutrosophic Theory and Its Applications in Algebra), Volume IX

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