Cosets in inverse semigroups and inverse subsemigroups of finite index

The index of a subgroup of a group counts the number of cosets of that subgroup. A subgroup of finite index often shares structural properties with the group, and the existence of a subgroup of finite index with some particular property can therefore imply useful structural information for the overgroup. Although a developed theory of cosets in inverse semigroups exists, it is defined only for closed inverse subsemigroups, and the structural correspondences between an inverse semigroup and a closed inverse subsemigroup of finte index are much weaker than in the group case. Nevertheless, many aspects of this theory remain of interest, and some of them are addressed in this thesis. We study the basic theory of cosets in inverse semigroups, including an index formula for chains of subgroups and an analogue of M. Hall’s Theorem on counting subgroups of finite index in finitely generated groups. We then look at specific examples, classifying the finite index inverse subsemigroups in polycyclic monoids and in graph inverse semigroups. Finally, we look at the connection between the properties of finite generation and having finte index: these were shown to be equivalent for free inverse monoids by Margolis and Meakin

Applications of Fuzzy Semiprimary Ideals under Group Action

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

Saturated Varieties of Semigroups

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

Characterization of Non-Linear Bi-Skew Jordan <i>n</i>-Derivations on Prime ∗-Algebras

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

Reversible Cyclic Codes over <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><mn>2</mn></msub><mo>+</mo><msub><mrow><mi>u</mi><mi mathvariant="double-struck">F</mi></mrow><mn>2</mn></msub><mo>+</mo><msub><mrow><msup><mi>u</mi><mn>2</mn></msup><mi mathvariant="double-struck">F</mi></mrow><mn>2</mn></msub></mrow></semantics></math></inline-formula>

In this article, we investigate the formation of reversible cyclic codes (i.e., its codewords forms a symmetry) over the ring S=F2+uF2+u2F2, where u3=0. We find a unique set of generators for cyclic codes over S and classify reversible cyclic codes to their generators. The dual reversible cyclic codes are studied as well. Moreover, we provide some examples of reversible cyclic codes

(<i>β</i>,<i>γ</i>)-Skew QC Codes with Derivation over a Semi-Local Ring

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 S[y;β,γ] becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of S[y;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

Centrally Extended Jordan (∗)-Derivations Centralizing Symmetric or Skew Elements

Let A be a non-commutative prime ring with involution ∗, of characteristic ≠2(and3), with Z as the center of A and Π a mapping Π:A→A such that [Π(x),x]∈Z for all (skew) symmetric elements x∈A. If Π is a non-zero CE-Jordan derivation of A, then A satisfies s4, the standard polynomial of degree 4. If Π is a non-zero CE-Jordan ∗-derivation of A, then A satisfies s4 or Π(y)=λ(y−y*) for all y∈A, and some λ∈C, the extended centroid of A. Furthermore, we give an example to demonstrate the importance of the restrictions put on the assumptions of our results

Construction of quantum codes over the class of commutative rings and their applications to DNA codes

WOS:000961861500001Let (Formula presented.) be positive integers and (Formula presented.) be a finite field of order (Formula presented.) of characteristic 2. The primary goal of this paper is to study the structural properties of cyclic codes over the ring (Formula presented.), for (Formula presented.), where (Formula presented.) is the non-zero element of (Formula presented.). As an application, we obtain better quantum error correcting codes over the ring (Formula presented.) (for (Formula presented.)). Moreover, we acquire optimal linear codes with the help of the Gray image of cyclic codes. Finally, we present methods for reversible DNA codes