G. Muhiuddin

On -Derivations of BCI-Algebras

We introduce the notion of t-derivation of a BCI-algebra and investigate related properties. Moreover, we study t-derivations in a p-semisimple BCI-algebra and establish some results on t-derivations in a p-semisimple BCI-algebra.

On -Derivations in BCI-Algebras

The notion of (regular) -derivations of a BCI-algebra is introduced, some useful examples are discussed, and related properties are investigated. The condition for a -derivation to be regular is provided. The concepts of a -invariant -derivation and -ideal are introduced, and their relations are discussed. Finally, some results on regular -derivations are obtained.

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.

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.

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.

On Symmetric Left Bi-Derivations in BCI-Algebras

The notion of symmetric left bi-derivation of a BCI-algebra X is introduced, and related properties are investigated. Some results on componentwise regular and d-regular symmetric left bi-derivations are obtained. Finally, characterizations of a p-semisimple BCI-algebra are explored, and it is proved that, in a p-semisimple BCI-algebra, F is a symmetric left bi-derivation if and only if it is a symmetric bi-derivation.

Hesitant fuzzy translations and extensions of subalgebras and ideals in BCK/ BCI-algebras

The first author would like to acknowledge financial support for this work, from the Deanship of Scientific Research (DRS), University of Tabuk, Tabuk, Saudi Arabia, under grant no. S/0123/1436.

Non-differentiable minimax fractional programming with higher-order type Ifunctions

In this article, we are concerned with a class of non-differentiable minimaxfractional programming problems and their higher-order dual model. Weak, strongand converse duality theorems are discussed involving generalized higher-ordertype I functions. The presented results extend some previously known results onnon-differentiable minimax fractional programming.

Normal unisoft filters in R0-algebras

In the present paper, the notions of normal uni-soft filters in -algebras are introduced, and related properties are investigated. Furthermore, characterizations of a normal uni-soft filter are established, and a new normal uni-soft filter from old one is constructed. Finally, a condition for a uni-soft filter to be normal is given and a condensational property of a normal uni-soft filter is established.

Hesitant fuzzy semigroups with a frontier

The notion of a hesitant fuzzy semigroup with a frontier is introduced, and several properties are investigated. Characterizations of a hesitant fuzzy semigroup with a frontier are considered, and a condition for a special set to be a subsemigroup is provided. The hesitant union and hesitant intersection of two hesitant fuzzy semigroups with a frontier are dealt with, and the hesitant fuzzy pre-image and hesitant fuzzy image of a hesitant fuzzy semigroup with a frontier under the homomorphism are discussed.