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

The notion of (α̃, β̃)-fuzzy left (right, bi-) ideals in semigroups is introduced, and related properties are investigated. Characterizations of (∈, ∈∨ qδ 0)-fuzzy left (right, bi-) ideals are provided. Mathematics Subject Classification: 20M12, 03E72, 08A72

A novel generalization of fuzzy ideals in semigroups

The notion of intuitionistic fuzzy a-ideals in BCI-algebras is introduced. Conditions for an intuitionistic fuzzy ideal to be an intuitionistic fuzzy a-ideal are provided. Using a collection of a-ideals, intuitionistic fuzzy a-ideals are established.

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Intuitionistic Fuzzifications of a-Ideals in BCI-Algebras

The notion of positive immplicative Smarandache BCC-ideals is indroduced, and related properties are investigated.

Positive Implicative Smarandache BCC-Ideals in Smarandache BCC-Algebras

The concept of a commutative soju ideal in a BCK-algebra and a BCI-algebra is introduced, and their properties are investigated. The relationship between a soju ideal and a commutative soju ideal are discussed, and examples to show that any soju ideal may not be a commutative soju ideal are provided. Conditions for a soju ideal to be a commutative soju ideal are considered, and characterizations of a commutative soju ideal are studied. A new commutative soju ideal using the given commutative soju ideal is maded, and the extension property for a commutative soju ideal is established. A commutative soju ideal is established by using a commutative ideal of a BCI-algebra. The notion of a closed soju ideal in a BCI-algebra is also introduced, and it is used in studying the characterization of a commutative soju ideal.

Commutative ideals of BCK-algebras and BCI-algebras based on soju structures

The concepts of (∈, ∈ ∨ qδ 0)-fuzzy subalgebras and ∈ ∨ qδ 0-level sets Corresponding author 7188 G. Muhiuddin, Seok Zun Song, Hee Sik Kim, Young Bae Jun are introduced, and related properties are investigated. Relations between an (∈,∈)-fuzzy subalgebra and an (∈, ∈ ∨ qδ 0)-fuzzy subalgebra are discussed, and characterizations of (∈, ∈∨ qδ 0)-fuzzy subalgebras are discussed. Homomorphic image and pre-image of an (∈, ∈ ∨ qδ 0)-fuzzy subalgebra are considered. Mathematics Subject Classification: 06F35, 03G25, 08A72

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Seok Zun Song

Papers

A novel generalization of fuzzy ideals in semigroups

Intuitionistic Fuzzifications of a-Ideals in BCI-Algebras

Positive Implicative Smarandache BCC-Ideals in Smarandache BCC-Algebras

Commutative ideals of BCK-algebras and BCI-algebras based on soju structures

Characterizations of Fuzzy Subalgebras in BCK=BCI-Algebras