Roughness of soft sets and fuzzy sets in semigroups based on set-valued picture hesitant fuzzy relations

<abstract><p>In the philosophy of rough set theory, the methodologies of rough soft sets and rough fuzzy sets have been being examined to be efficient mathematical tools to deal with unpredictability. The basic of approximations in rough set theory is based on equivalence relations. In the aftermath, such theory is extended by arbitrary binary relations and fuzzy relations for more wide approximation spaces. In recent years, the notion of picture hesitant fuzzy relations by Mathew et al. can be considered as a novel extension of fuzzy relations. Then this paper proposes extended approximations into rough soft sets and rough fuzzy sets from the viewpoint of its. We give corresponding examples to illustrate the correctness of such approximations. The relationships between the set-valued picture hesitant fuzzy relations with the upper (resp., lower) rough approximations of soft sets and fuzzy sets are investigated. Especially, it is shown that every non-rough soft set and non-rough fuzzy set can be induced by set-valued picture hesitant fuzzy reflexive relations and set-valued picture hesitant fuzzy antisymmetric relations. By processing the approximations and advantages in the new existing tools, some terms and products have been applied to semigroups. Then, we provide attractive results of upper (resp., lower) rough approximations of prime idealistic soft semigroups over semigroups and fuzzy prime ideals of semigroups induced by set-valued picture hesitant fuzzy relations on semigroups.</p></abstract>

Generalizations of S- Prime Ideals

Let R be a commutative ring with identity and S be a multiplicative subset of R . In this paper we introduce the concept of almost S-prime ideal as a new generalization of S−prime ideal. Let P be a proper ideal of R disjoint with S. Then P is said to be almost S- prime ideal if there exists s ∈ S such that, for all x, y ∈ R if xy ∈ P − P 2 then sx ∈ P or sy ∈ P. Number of results concerning this concept and examples are given. Furthermore, we investigate an almost S- prime ideals of trivial ring extensions and amalgamation rings..

On 2-absorbing ideals of commutative semirings

Let [Formula: see text] be a commutative semiring with unity different than zero. In this paper, we study the concept of [Formula: see text]-absorbing ideals of [Formula: see text] which can be considered as a generalization of prime ideals. Among others, it is shown that the radical of a [Formula: see text]-absorbing ideal is also a [Formula: see text]-absorbing ideal and there are at most [Formula: see text] prime [Formula: see text]-ideals of [Formula: see text] that are minimal over a [Formula: see text]-absorbing ideals.

Finitary ideals of direct products in quantales

The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, appears in various areasof mathematics-in quantaloid theory, in non classical logic as completion of the Lindebaum algebra, and in different representations of the spectrum of a C∗ algebra asmany-valued and non commutative topologies. To put it briefly, its importance is nolonger to be demonstrated. Quantales are ring-like structures in that they share withrings the common fact that while as rings are semi groups in the tensor category ofabelian groups, so quantales are semi groups in the tensor category of sup-lattices.In 2008 Anderson and Kintzinger [1] investigated the ideals, prime ideals, radical ideals, primary ideals, and maximal of a product ring R × S of two commutativenon non necceray unital rings R and S: Something resembling rings are quantales byanalogy with what is studied in ring, we begin an investigation on ideals of a productof two quantales. In this paper, given two quantales Q1 and Q2; not necessarily withidentity, we investigate the ideals, prime ideals, primary ideals, and maximal ideals ofthe quantale Q1 × Q2

Graded 1-absorbing prime ideals of graded commutative rings

Let [Formula: see text] be a group with identity [Formula: see text] and [Formula: see text] be [Formula: see text]-graded commutative ring with [Formula: see text] In this paper, we introduce and study the graded versions of 1-absorbing prime ideal. We give some properties and characterizations of these ideals in graded ring, and we give a characterization of graded 1-absorbing ideal the idealization [Formula: see text]

A Density of Ramified Primes

Let K be a cyclic number field of odd degree over $${\mathbb {Q}}$$ Q with odd narrow class number, such that 2 is inert in $$K/{\mathbb {Q}}$$ K / Q . We define a family of number fields $$\{K(p)\}_p$$ { K ( p ) } p , depending on K and indexed by the rational primes p that split completely in $$K/{\mathbb {Q}}$$ K / Q , in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in $$K(p)/{\mathbb {Q}}$$ K ( p ) / Q is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension $$K/{\mathbb {Q}}$$ K / Q . Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.