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

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

Innovative types of fuzzy gamma ideals in ordered gamma semigroups

The fuzzification of algebraic structures plays an important role in handling many areas of multi-disciplinary research, such as computer science, control theory, information science, topological spaces and fuzzy automata to handle many real world problems. For instance, algebraic structures are particularly useful in detecting permanent faults on sequential machine behaviour. However, the idea of ordered T-semigroup as a generalization of ordered semigroup in algebraic structures has rarely been studied. In this research, a new form of fuzzy subsystem in ordered T-semigroup is defined. Specifically, a developmental platform of further characterizations on ordered T-semigroups using fuzzy subsystems properties and new fuzzified ideal structures of ordered semigroups is developed based on a detailed study of ordered T-semigroups in terms of the idea of belongs to (E) and quasicoincidence with (q) relation. This idea of quasi-coincidence of a fuzzy point with a fuzzy set played a remarkable role in obtaining several types of fuzzy subgroups and subsystems based on three contributions. One, a new form of generalization of fuzzy generalized bi T-ideal is developed, and the notion of fuzzy bi T-ideal of the form (E,E Vqk) in an ordered T-semigroup is also introduced. In addition, a necessary and sufficient condition for an ordered T-semigroup to be simple T-ideals in terms of this new form is stated. Two, the concept of (E,E Vqk)-fuzzy quasi T-ideals, fuzzy semiprime T-ideals, and other characterization in terms of regular (left, right, completely, intra) in ordered T-semigroup are developed. Three, a new fuzzified T-ideal in terms of interior T-ideal of ordered T-semigroups in many classes are determined. Thus, this thesis provides the characterizations of innovative types of fuzzy T-ideals in ordered T-semigroups with classifications in terms of completely regular, intra-regular, for fuzzy generalized bi T-ideals, fuzzy bi T-ideals, fuzzy quasi and fuzzy semiprime T-ideals, and fuzzy interior T-ideals. These findings constitute a platform for further advancement of ordered T-semigroups and their applications to other concepts and branches of algebra

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

Rough U-Exact Sequence of Rough Groups

The notion of a U-exact sequence is a generalization of the exact sequence. In this paper, we introduce a rough U-exact sequence in a rough group in an approximation space. Furthermore, we provide the properties of the rough U-exact sequence in a rough group

LL-fuzzy ideal degrees in effect algebras

summary:In this paper, considering $L$ being a completely distributive lattice, we first introduce the concept of $L$-fuzzy ideal degrees in an effect algebra $E$, in symbol $\mathfrak{D}_{ei}$. Further, we characterize $L$-fuzzy ideal degrees by cut sets. Then it is shown that an $L$-fuzzy subset $A$ in $E$ is an $L$-fuzzy ideal if and only if $\mathfrak{D}_{ei}(A)=\top,$ which can be seen as a generalization of fuzzy ideals. Later, we discuss the relations between $L$-fuzzy ideals and cut sets ($L_{\beta}$-nested sets and $L_{\alpha}$-nested sets). Finally, we obtain that the $L$-fuzzy ideal degree is an $(L,L)$-fuzzy convexity. The morphism between two effect algebras is an $(L,L)$-fuzzy convexity-preserving mapping

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