Effects of the material properties and thickness on the minimum formable radius in the subtle feature forming process.

The subtle feature geometry also called a feature line, is considered an important geometric characteristic of automotive outer panels. The influences of material properties and thickness on the radius of curvature of subtle features were investigated in this study. First, the stamping process was simplified to a combined forming process between tensile and bending deformation. Subsequently, test materials, namely, 180B2, 210B2, CR2, CR3, and CR4, with various thickness values were adopted in the finite element analysis and experiments. In addition, the radius of curvature with respect to the material, thickness, punch radius, and punch angle was studied. The simulation results were compared with the experimental results for verification. From this comparison, it was found that the simulation results were in good agreement with the experimental data. Finally, the forming characteristics of the subtle feature-forming process were investigated to determine the effects of the material properties and thickness on the radius of curvature. The reason for the minimum formable radius when the radius of the punch was zero was studied. The results showed that, as the material thickness increased, more concentrated deformation occurred in the central region. In contrast, the radius of curvature of the subtle features increased as the thickness of the central region decreased. Similarly, decreased n-value results were identified for the same reason as the increased radius of curvature.

Int-soft (generalized) bi-ideals of semigroups.

The notions of int-soft semigroups and int-soft left (resp., right) ideals in semigroups are studied in the paper by Song et al. (2014). In this paper, further properties and characterizations of int-soft left (right) ideals are studied, and the notion of int-soft (generalized) bi-ideals is introduced. Relations between int-soft generalized bi-ideals and int-soft semigroups are discussed, and characterizations of (int-soft) generalized bi-ideals and int-soft bi-ideals are considered. Given a soft set (α;S) over U, int-soft (generalized) bi-ideals generated by (α;S) are established.

Concave soft sets, critical soft points, and union-soft ideals of ordered semigroups.

The notions of union-soft semigroups, union-soft l-ideals, and union-soft r-ideals are introduced, and related properties are investigated. Characterizations of a union-soft semigroup, a union-soft l-ideal, and a union-soft r-ideal are provided. The concepts of union-soft products and union-soft semiprime soft sets are introduced, and their properties related to union-soft l-ideals and union-soft r-ideals are investigated. Using the notions of union-soft l-ideals and union-soft r-ideals, conditions for an ordered semigroup to be regular are considered. The concepts of concave soft sets and critical soft points are introduced, and their properties are discussed.

Hesitant fuzzy soft subalgebras and ideals in BCK/BCI-algebras.

As a link between classical soft sets and hesitant fuzzy sets, the notion of hesitant fuzzy soft sets is introduced and applied to a decision making problem in the papers by Babitha and John (2013) and Wang et al. (2014). The aim of this paper is to apply hesitant fuzzy soft set for dealing with several kinds of theories in BCK/BCI-algebras. The notions of hesitant fuzzy soft subalgebras and (closed) hesitant fuzzy soft ideals are introduced, and related properties are investigated. Relations between a hesitant fuzzy soft subalgebra and a (closed) hesitant fuzzy soft ideal are discussed. Conditions for a hesitant fuzzy soft set to be a hesitant fuzzy soft subalgebra are given, and conditions for a hesitant fuzzy soft subalgebra to be a hesitant fuzzy soft ideal are provided. Characterizations of a (closed) hesitant fuzzy soft ideal are considered.

Classes of int-soft filters in residuated lattices.

The notions of int-soft filters, int-soft G-filters, regular int-soft filters, and MV-int-soft filters in residuated lattices are introduced, and their relations, properties, and characterizations are investigated. Conditions for an int-soft filter to be an int-soft G-filter, a regular int-soft filter, or an MV-int-soft filter are provided. The extension property for an int-soft G-filter is discussed. Finally, it is shown that the notion of an MV-int-soft filter coincides with the notion of a regular int-soft filter in BL-algebras.

Ideal theory in semigroups based on intersectional soft sets.

The notions of int-soft semigroups and int-soft left (resp., right) ideals are introduced, and several properties are investigated. Using these notions and the notion of inclusive set, characterizations of subsemigroups and left (resp., right) ideals are considered. Using the notion of int-soft products, characterizations of int-soft semigroups and int-soft left (resp., right) ideals are discussed. We prove that the soft intersection of int-soft left (resp., right) ideals (resp., int-soft semigroups) is also int-soft left (resp., right) ideals (resp., int-soft semigroups). The concept of int-soft quasi-ideals is also introduced, and characterization of a regular semigroup is discussed.

Subalgebras of BCK/BCI-algebras based on cubic soft sets.

Operations of cubic soft sets including "AND" operation and "OR" operation based on P-orders and R-orders are introduced and some related properties are investigated. An example is presented to show that the R-union of two internal cubic soft sets might not be internal. A sufficient condition is provided, which ensure that the R-union of two internal cubic soft sets is also internal. Moreover, some properties of cubic soft subalgebras of BCK/BCI-algebras based on a given parameter are discussed.

Decomposition of fuzzy soft sets with finite value spaces.

The notion of fuzzy soft sets is a hybrid soft computing model that integrates both gradualness and parameterization methods in harmony to deal with uncertainty. The decomposition of fuzzy soft sets is of great importance in both theory and practical applications with regard to decision making under uncertainty. This study aims to explore decomposition of fuzzy soft sets with finite value spaces. Scalar uni-product and int-product operations of fuzzy soft sets are introduced and some related properties are investigated. Using t-level soft sets, we define level equivalent relations and show that the quotient structure of the unit interval induced by level equivalent relations is isomorphic to the lattice consisting of all t-level soft sets of a given fuzzy soft set. We also introduce the concepts of crucial threshold values and complete threshold sets. Finally, some decomposition theorems for fuzzy soft sets with finite value spaces are established, illustrated by an example concerning the classification and rating of multimedia cell phones. The obtained results extend some classical decomposition theorems of fuzzy sets, since every fuzzy set can be viewed as a fuzzy soft set with a single parameter.

Characterizations of ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals.

In this paper, we give characterizations of ordered semigroups in terms of (∈, ∈ âˆ¨q)-fuzzy interior ideals. We characterize different classes regular (resp. intra-regular, simple and semisimple) ordered semigroups in terms of (∈, ∈ âˆ¨q)-fuzzy interior ideals (resp. (∈, ∈ âˆ¨q)-fuzzy ideals). In this regard, we prove that in regular (resp. intra-regular and semisimple) ordered semigroups the concept of (∈, ∈ âˆ¨q)-fuzzy ideals and (∈, ∈ âˆ¨q)-fuzzy interior ideals coincide. We prove that an ordered semigroup S is simple if and only if it is (∈, ∈ âˆ¨q)-fuzzy simple. We characterize intra-regular (resp. semisimple) ordered semigroups in terms of (∈, ∈ âˆ¨q)-fuzzy ideals (resp. (∈, ∈ âˆ¨q)-fuzzy interior ideals). Finally, we consider the concept of implication-based fuzzy interior ideals in an ordered semigroup, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.