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.

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In this article, we have designed a new scheme for the construction of the nonlinear confusion component. Our mechanism uses the notion of a semigroup, Inverse LA-semigroup, and various other loops. With the help of these mathematical structures, we can easily build our confusion component namely substitution boxes (S-boxes) without having specialized structures. We authenticate our proposed methodology by incorporating the available cryptographic benchmarks. Moreover, we have utilized the technique for order of preference by similarity to ideal solution (TOPSIS) to select the best nonlinear confusion component. With the aid of this multi-criteria decision-making (MCDM), one can easily select the best possible confusion component while selecting among various available nonlinear confusion components.

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.

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.