Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we initiate the study of soft semirings by using the soft set theory. The notions of soft semirings, soft subsemirings, soft ideals, idealistic soft semirings and soft semiring homomorphisms are introduced, and several related properties are investigated.

Soft semirings

Molodtsov [D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19-31] 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 the theory of BCK/BCI-algebras. The notion of soft BCK/BCI-algebras and soft subalgebras are introduced, and their basic properties are derived.

Soft BCK/BCI-algebras

Applications of soft sets in ideal theory of BCK/BCI-algebras

Graphical abstractDisplay Omitted In this paper, we investigate the relationships among rough sets, soft sets and hemirings. The concept of soft rough hemirings is introduced, which is an extended notion of a rough hemiring. It is pointed out that in this paper, we first apply soft rough sets to algebraic structure-hemirings. Further, we first put forward the concepts of C-soft sets and CC-soft sets, which provide a new research idea for soft rough algebraic research. Moreover, we study roughness in hemirings with respect to MSR-approximation spaces. Some new soft rough operations over hemirings are explored. In particular, lower and upper MSR-hemirings (k-ideal and h-ideal) are investigated. Finally, we put forth an approach for multicriteria group decision making problem based on modified soft rough sets and offer an actual example.

A novel soft rough set

Molodtsov [D. Molodtsov, Soft set theory-First results, Comput. Math. Appl. 37 (1999) 19-31] 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. Jun [Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408-1413] applied first the notion of soft sets by Molodtsov to the theory of BCK/BCI-algebras. In this paper we introduce the notion of soft p-ideals and p-idealistic soft BCI-algebras, and then investigate their basic properties. Using soft sets, we give characterizations of (fuzzy) p-ideals in BCI-algebras. We provide relations between fuzzy p-ideals and p-idealistic soft BCI-algebras.

Soft p-ideals of soft BCI-algebras

Generalized fuzzy interior ideals in semigroups

On fuzzy h -ideals in hemirings

Codes based on BCK-algebras

The notions of bipolar fuzzy subalgebras and bipolar fuzzy closed ideals in BCH-algebras are introduced, and related properties are investigated. Relations between a bipolar fuzzy subalgebra and a bipolar fuzzy closed ideal are considered. Conditions for a bipolar fuzzy subalgebra to be a bipolar fuzzy closed ideal are provided. Characterizations of a bipolar fuzzy closed ideal are discussed.

Subalgebras and closed ideals of bch-algebras based on bipolar-valued fuzzy sets

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.

Seok Zun Song

Papers

Generalized fuzzy interior ideals in semigroups

On fuzzy h -ideals in hemirings

Codes based on BCK-algebras

Subalgebras and closed ideals of bch-algebras based on bipolar-valued fuzzy sets

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