In this paper, we acquaint new kinds of ideals of BCK-algebras built on tripolar picture fuzzy structures. In fact, the notions of tripolar picture fuzzy ideal, tripolar picture fuzzy implicative ideal (commutative ideal) of BCK-algebra are introduced, and related properties are studied. Also, a relation among tripolar picture fuzzy ideal, and tripolar picture fuzzy implicative ideal is well-known. Furthermore, it is shown that a tripolar picture fuzzy implicative ideal of BCK-algebra may be a tripolar picture fuzzy ideal, but the converse is not correct in common. Further, it is obtained that in an implicative BCK-algebra, the converse of aforementioned statement is true. Finally, the opinion of tripolar picture fuzzy commutative ideal is given, and some useful properties are explored. Many examples are constructed to sport our study.

The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object’s symmetries. In this paper, we introduce new concepts in an algebraic structure called BCI-algebra, where we present the concepts of bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy (closed) BCI-commutative ideals of BCI-algebras. The relationship between bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy ideals is investigated, and various conditions are provided for a bipolar fuzzy ideal to be a bipolar fuzzy BCI-positive implicative ideal. Furthermore, conditions are presented for a bipolar fuzzy (closed) ideal to be a bipolar fuzzy BCI-commutative ideal.

We give an accessible introduction into the theory of lower central series of associative algebras, exhibiting the interplay between algebra, geometry and representation theory that is characteristic for this subject, and discuss some open questions. In particular, we provide shorter and clearer proofs of the main results of this theory. We also discuss some new theoretical and computational results and conjectures on the lower central series of the free algebra in two generators modulo a generic homogeneous relation.

In this study, by generalizing the notion of fuzzy bi-ideals of ordered semirings, the notion of $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals is established. We prove that $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals are fuzzy bi-ideals but that the converse is not true, and an example is provided to support this proof. A condition is given under which fuzzy bi-ideals of ordered semirings coincide with $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals. An equivalent condition and certain correspondences between bi-ideals and $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals are presented. Moreover, the $({\kappa }^{*},\kappa )$-lower part of $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-ideals is described and depicted in terms of several classes of ordered semirings. Furthermore, it is shown that the ordered semiring is bi-simple if and only if it is $(\in ,\in \vee ({\kappa }^{*},q_{\kappa }))$-fuzzy bi-simple.

In this paper, we study the connection between topological spaces, hyperrings (semi-hypergroups), and rough sets. We concentrate here on the topological parts of the lower and upper approximations of hyperideals in hyperrings and semi-hypergroups. We provide the conditions for the boundary of hyp-ideals of a hyp-ring to become the hyp-ideals of hyp-ring.

This paper concerns three relationship between the recently proposed cubic sets and finite state machines. The notions of cubic finite state machine (cubic FSM), a subsystem of cubic FSM and cartesian composition (direct product, P-(R-) union, and P-(R-) intersection) of two subsystems of cubic FSMs are introduced. We study the cartesian composition, direct product and union of two subsystems of cubic FSMs is a subsystem of a cubic FSM. We provide many examples on each case. We consider conditions for subsystem of cubic FSM to be both an internal cubic subsystem of cubic FSM and an external cubic subsystem of cubic FSM.