In this paper, we apply the concept of linear Diophantine fuzzy sets in $BCK/BCI$-algebras. In this respect, the notions of linear Diophantine fuzzy subalgebras and linear Diophantine fuzzy (commutative) ideals are introduced and some vital properties are discussed. Additionally, characterizations of linear Diophantine fuzzy subalgebras and linear Diophantine fuzzy (commutative) ideals are considered. Moreover, the associated results for linear Diophantine fuzzy subalgebras, linear Diophantine fuzzy ideals and linear Diophantine fuzzy commutative ideals are obtained.

The notion of a linear diophantine fuzzy set as a generalization of a fuzzy set is a mathematical approach that deals with vagueness in decision-making problems. The use of reference parameters associated with validity and non-validity functions in linear diophantine fuzzy sets makes it more applicable to model vagueness in many real-life problems. On the other hand, subspaces of vector spaces are of great importance in many fields of science. The aim of this paper is to combine the two notions. In this regard, we consider the linear diophantine fuzzification of a vector space by introducing and studying the linear diophantine fuzzy subspaces of a vector space. First, we studied the behaviors of linear diophantine fuzzy subspaces of a vector space under a linear diophantine fuzzy set. Second, and by means of the level sets, we found a relationship between the linear diophantine fuzzy subspaces of a vector space and the subspaces of a vector space. Finally, we discuss the linear diophantine fuzzy subspaces of a quotient vector space.

Let G be a group and S be a subset of G such that e ∉ S and S−1 ⊆ S. Then Cay(G, S) is a simple undirected Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy−1 ∈ S. The size of subset S is called the valency of Cay(G, S). In this paper, we determined the structure of all Cay(D2n, S), where D2n is a dihedral group of order 2n, n ≥ 3 and |S| = 1, 2 or 3.

Let G be a group. In this paper, we introduce and study the intersection subgroup graph of G as the (simple) graph ${\Gamma }_S(G)$ with vertices the nontrivial subgroups H of G such that $H\cap K=\left\{e\right\}$ for some nontrivial subgroup K of G, and distinct vertices H and K are adjacent if and only if $H\cap K=\left\{e\right\}$. We investigate when ${\Gamma }_S(G)$ is a nonempty, finite, complete, complete bipartite, or planar graph.

Algebraic structure consisting of a set together with an associative internal binary operation on it, so called semigroup has applications in different fields of science. For a better understanding of these applications, semigroups are characterized through their subsets. Fuzzy sets deal with uncertainties, and because many real-life problems have an associated algebraic structure, fuzzification of these structures makes sense and is useful. In this paper, our aim is to characterize a semigroup through a generalized concept of its bi-antiideals and to fuzzify this concept. In particular, by using fuzzy sets, we studied fuzzy $(m,n)$-bi-antiideals of a semigroup and linked it to $(m,n)$-bi-antiideals by mean of level sets.