An approach to Q-neutrosophic soft rings

In this paper, the concept of neutrosophic soft ring is introduced and some basic properties related to it are established. Then the Cartesian product of neutrosophic soft ring, neutrosophic soft subring, neutrosophic normal soft ring, neutrosophic soft ideal are defined and illustrated by suitable examples. Several related properties and structural characteristics of these are studied here. Some theorems are also developed in each part.

In this article, we apply the concept of a t -intuitionistic fuzzy set to PMS-ideals in PMS-algebras. The notion of the t -intuitionistic fuzzy PMS-ideal of PMS-algebra is introduced, and several related properties are studied. The relationships between a t -intuitionistic fuzzy PMS-ideal and a t -intuitionistic fuzzy PMS-subalgebra of a PMS-algebra, as well as the relationships between an intuitionistic fuzzy PMS-ideal and a t -intuitionistic fuzzy PMS-ideal are discussed in detail. A condition for an intuitionistic fuzzy set to be a t -intuitionistic fuzzy PMS-ideal is provided. The t -intuitionistic fuzzy PMS-ideals of PMS-algebra are described using their α , β level cuts. The homomorphism of a t -intuitionistic fuzzy PMS-ideal of a PMS-algebra is studied, and its homomorphic image and inverse image are explored. The Cartesian product of any two t -intuitionistic fuzzy PMS-ideals is discussed, and some related results are derived. The Cartesian product of the t -intuitionistic fuzzy PMS-ideals is also characterized using its α , β level cuts. The strongest t -intuitionistic fuzzy PMS-relation in a PMS-algebra is defined. Finally, the relationships between the strongest t -intuitionistic fuzzy PMS-relation and t -intuitionistic fuzzy PMS-ideal are studied.

In this paper, we apply the concept of bipolar fuzzy sets to pseudo-UP ideals in pseudo-UP algebras. We prove that the intersection of two bipolar fuzzy pseudo-UP ideals is also a bipolar fuzzy pseudo-UP ideal, while the union of two such ideals does not always result in a bipolar fuzzy pseudo-UP ideal. Additionally, we discuss the concepts of bipolar fuzzy pseudo-UP ideals under homomorphism and explore several related properties. The homomorphic image and inverse image of bipolar fuzzy pseudo-UP ideals in a pseudo-UP algebra are also examined in detail. Furthermore, we study the notion of a bipolar fuzzy pseudo-UP ideal under the Cartesian product of pseudo-UP algebra. The Cartesian product of any two bipolar fuzzy pseudo-UP ideals is also the bipolar fuzzy pseudo-UP ideal of pseudo-UP algebra, and then some related results are obtained. MSC: 03G25, 06D30.

The concept of soft translations of soft subalgebras and soft ideals over BCI/BCK-algebras is introduced and some related properties are studied. Notions of Soft extensions of soft subalgebras and soft ideals over BCI/BCK-algebras are also initiated. Relationships between soft translations and soft extensions are explored.

Molodtsov (5) introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to an implicative ideal of BCK-algebras. The notion of implicative soft ideals in BCK- algebras and implicative idealistic soft BCK-algebras is introduced, and related properties are investigated. Relations between implicative soft ideals and commutative (resp. positive implicative) soft ideals are dis- cussed. Also, relations between implicative idealistic soft BCK-algebras and commutative (resp. positive implicative) idealistic soft BCK-algebras are provided.

Soft semirings

In this paper, we introduce a generalization of the well-known notion of a Noetherian ring which we call a Q-Noetherian ring. We investigate the transfer of the Q-Noetherian property to trivial ring extensions, localizations, homomorphic image of rings, direct product of rings, and amalgamated algebras along an ideal. These results provide examples of non-Noetherian Q-Noetherian rings, of non-coherent Q-Noetherian rings and examples of non-Q-Noetherian coherent rings. The article includes a brief discussion of the scope and precision of our results.

In this paper, the notion of cubic subalgebras of BCH-algebras are introduced. Some characterization of cubic subalgebras of BCH-algebras are given. The homomorphic image and inverse image of cubic subalgebras are studied and investigated some related properties

It is shown that each commutative Artin local ring having each of its ideals generated by two elements is the homomorphic image of a one-dimensional local complete intersection ring which also has each of its ideals generated by two elements. It is indicated how this can be applied to show that the property that each ideal is projective over its endomorphism ring does not pass to homomorphic images, and in determining the commutative group rings with the two-generator property.

In this paper, the notion of picture fuzzy subspace of a crisp vector space is established and some related properties are explored on the basis of some basic operations (intersection, Cartesian product, union, (θ, ϕ, ψ)-cut etc.) on picture fuzzy sets. Direct sum of two picture fuzzy subspaces is initiated here over the direct sum of two crisp vector spaces. Also, the concepts of picture fuzzy linear transformation and picture fuzzy linearly independent set of vectors are introduced and some corresponding results are presented. Isomorphism between two picture fuzzy subspaces is developed here as an extension of isomorphism between two fuzzy subspaces

Emerging trends in soft set theory and related topics.

If we have two arbitrary non empty sets ,then their cartesian product can be constructed. Cartesian products of two sets can be generalized into number of sets. It has been found that if the algebraic structure of groups and rings are seen as any set, then the phenomenon of cartesian products of sets can be extended to groups and rings. Direct products of groups and rings can be obtained by adding binary operations to the cartesian product. This paper answers the question of whether the direct product phenomenon of groups and rings can also be extended at the near ring and Smarandache near ring ?. The method in this paper is by following the method in groups and rings, namely by seen that near ring and Smarandache near ring as a set and then build their cartesian products. Next, the binary operations is adding to the cartesian products that have been obtained to build the direct product definitions of near ring and near ring Smarandache.

The concept and properties of cubic sets were first introduced and investigated by Jun et al. in 2011. In this paper, we first introduce the concept of cubic sets in UP-subalgebras and consider the UP-ideals of a UP-algebra. Relationships between the cubic UP-subalgebras and the cubic UP-ideals of a UP-algebra are investigated. In addition, the homomorphic images and the inverse images of the cubic UP-subalgebras and the cubic UP-ideals are also studied. Related properties of UP -algebras and their UP-ideals are explored and discussed. Finally, the Cartesian product of cubic UP-algebras are investigated.

Derived strengths of preference relations on coordinates

In this paper, we first introduce the concept of quasi-coincidence of an intuitionistic fuzzy point within an intuition- istic fuzzy set. By using this new idea, we further introduce the notions of (∈, ∈∨ q)-intuitionistic fuzzy BCI-subalgebras of BCI-algebras and investigate some of their related properties. Some characterization theorems of these generalized intuitionis- tic fuzzy BCI-subalgebras are derived. Then we study the cartesian product of (∈, ∈∨ q)-intuitionistic fuzzy BCI-subalgebras. Finally, we introduce the homomorphism of intuitionistic fuzzy BCI-algebras, and prove that image and inverse image of an intuitionistic fuzzy set on a BCI-algebra should be an (∈, ∈∨ q)-intuitionistic fuzzy BCI-subalgebra on the basis of homomorphism of BCI-algebra.