This study delves into the structure and properties of left inverse zero divisor bands within semigroups, identifying their maximal forms and broadening the theoretical landscape of semigroup ...analysis. A significant focus is placed on the automorphisms of a semigroup S of centralizers of idempotent transformations with restricted range, revealing that these automorphisms are inner ones and induced by the units of S. Additionally, we establish that the automorphism group Aut(S) is isomorphic to US, the group of units of S. These findings extend previous results on semigroups of transformations, enhancing their applicability and providing a more unified theory. The practical implications of this work span multiple fields, including automata theory, coding theory, cryptography, and graph theory, offering tools for more efficient algorithms and models. By simplifying complex concepts and providing a solid foundation for future research, this study makes significant contributions to both theoretical and applied mathematics.
Saturated Varieties of Semigroups Nabi, Muneer; Alali, Amal S.; Bano, Sakeena
Symmetry (Basel),
08/2023, Volume:
15, Issue:
8
Journal Article
Peer reviewed
Open access
The complete characterization of saturated varieties of semigroups remains an unsolved problem. The primary objective of this paper is to make significant progress in this direction. We initially ...demonstrate that the variety of semigroups defined by the identity axy=ayxa is saturated. The next main result establishes that the variety of semigroups determined by the identity axy=ayax is saturated. Finally, we show that medial semigroups satisfying the identity xy=xyn, where n≥2, are also saturated. These results collectively lead to the conclusion that epis from these saturated varieties are onto. This paper thus offers substantial progress towards the comprehensive characterization of saturated varieties of semigroups.
Let n≥2 be a fixed integer and A be a C∗-algebra. A permuting n-linear map G:An→A is known to be symmetric generalized n-derivation if there exists a symmetric n-derivation D:An→A such that ...Gς1,ς2,…,ςiςi′,…,ςn=Gς1,ς2,…,ςi,…,ςnςi′+ςiD(ς1,ς2,…,ςi′,…,ςn) holds ∀ςi,ςi′∈A. In this paper, we investigate the structure of C∗-algebras involving generalized linear n-derivations. Moreover, we describe the forms of traces of linear n-derivations satisfying certain functional identity.
Group actions are a valuable tool for investigating the symmetry and automorphism features of rings. The concept of fuzzy ideals in rings has been expanded with the introduction of fuzzy primary, ...weak primary, and semiprimary ideals. This paper explores the existence of fuzzy ideals that are semiprimary but neither weak primary nor primary. Furthermore, it defines a group action on a fuzzy ideal and examines the properties of fuzzy ideals and their level cuts under this group action. In fact, it aims to investigate the relationship between fuzzy semiprimary ideals and the radical of fuzzy ideals under group action. Additionally, it includes the results related to the radical of fuzzy ideals and fuzzy G-semiprimary ideals. Moreover, the preservation of the image and inverse image of a fuzzy G-semiprimary ideal of a ring R under certain conditions is also studied. It delves into the algebraic nature of fuzzy ideals and the radical under G-homomorphism of fuzzy ideals.
Let A be a prime *-algebra. A product defined as U•V=UV∗+VU∗ for any U,V∈A, is called a bi-skew Jordan product. A map ξ:A→A, defined as ξpnU1,U2,⋯,Un=∑k=1npnU1,U2,...,Uk−1,ξ(Uk),Uk+1,⋯,Un for all ...U1,U2,...,Un∈A, is called a non-linear bi-skew Jordan n-derivation. In this article, it is shown that ξ is an additive ∗-derivation.
The aim of this paper is to determine several saturated classes of structurally regular semigroups. First, we show that structurally (n,m)-regular semigroups are saturated in a subclass of semigroups ...for any pair (n,m) of positive integers. We also demonstrate that, for all positive integers n and k with 1≤k≤n, the variety of structurally (0,n)-left seminormal bands is saturated in the variety of structurally (0,k)-bands. As a result, in the category of structurally (0,k)-bands, epis from structurally (0,n)-left seminormal bands is onto.
Data security and privacy are considered to be the biggest problems faced by service providers who have worked with public data for a long time. A key element of modern encryption that is utilized to ...increase textual confusion is the Substitution box (S-box) and the algebraic strength of the S-box has a significant impact on how secure the encryption method is. In this article, we present a unique method that uses a linear fractional transformation on a finite field to produce cryptographically robust S-boxes. Firstly, we choose a specific irreducible polynomial of degree 8 in Z2x to construct GF(28). Later, we used the action of PGL(2,GF(28)) on GF(28) to generate a robust S-box. The effectiveness of the built-in S-box was evaluated using several criteria including non-linearity, differential uniformity, strict avalanche criteria, linear approximation probability, and bit independence criterion. The proposed S-box's characteristics are compared to those of most recent S-boxes to confirm the higher performance. Additionally, the S box was used to encrypt images to show its usefulness for multimedia security applications. We performed several tests, including contrast, correlation, homogeneity, entropy, and energy, to evaluate the success of the encryption technique. The proposed method for ciphering an image is very effective, as proven by its comparison with several S boxes.
Elliptic curve cryptography has gained attention due to its strong resilience against current cryptanalysis methods. Inspired by the increasing demand for reliable and secure cryptographic methods, ...our research investigates the relationship between complex mathematical structures and image encryption. A substitution box (S-box) is the single non-linear component of several well-known security systems. Mordell elliptic curves are used because of their special characteristics and the immense computational capacity of Galois fields. These S-boxes are dynamic, which adds a layer of complexity that raises the encryption process’s security considerably. We suggest an effective technique for creating S-boxes based on a class of elliptic curves over GF(2n),n≥8. We demonstrate our approach’s robustness against a range of cryptographic threats through thorough examination, highlighting its practical applicability. The assessment of resistance of the newly generated S-box to common attack methods including linear, differential, and algebraic attacks involves a thorough analysis. This analysis is conducted by quantifying various metrics such as non-linearity, linear approximation, strict avalanche, bit independence, and differential approximation to gauge the S-box’s robustness against these attacks. A recommended method for image encryption involves the use of built-in S-boxes to quickly perform pixel replacement and shuffling. To evaluate the efficiency of the proposed strategy, we employed various tests. The research holds relevance as it can provide alternative guidelines for image encryption, which could have wider consequences for the area of cryptography as a whole. We believe that our findings will contribute to the development of secure communication and data protection, as digital security is becoming increasingly important.
In this article, we consider a semi-local ring S=Fq+uFq, where u2=u, q=ps and p is a prime number. We define a multiplication yb=β(b)y+γ(b), where β is an automorphism and γ is a β-derivation on S so ...that Sy;β,γ becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of Sy;β,γ and obtain the most striking results that are better than previous findings. We also determine the structural properties of 1-generator skew cyclic and skew-quasi cyclic codes. Further, We demonstrate remarkable results of the above-mentioned codes over S. Finally, we find the duality of skew cyclic and skew-quasi cyclic codes using a symmetric inner product. These codes are further generalized to double skew cyclic and skew quasi cyclic codes and a table of optimal codes is calculated by MAGMA software.
Let n and m be fixed positive integers. In this paper, we establish some structural properties of prime rings equipped with higher derivations. Motivated by the works of Herstein and Bell-Daif, we ...characterize rings with higher derivations D=dii∈N satisfying (i) dnx,dmy∈ZR for all x,y∈R and (ii) dnx,y∈ZR for all x,y∈R.
