Algebraic structures and their applications
Year:2022 | Volume:9 | Issue:1|Papers Count:12

In this paper, we use the Key-Moori Method 1 and construct a quaternary code C8 from a primitive representation of the group PSL2(9) of degree 15. We see that C8 is a self-orthogonal even code with the automorphism group isomorphic to the alternating group A8...

The concept of very true GE-algebra using very true operator is introduced and its properties are studied to expand the scope of research of GE-algebras. The concepts of simple very true GE-algebra and very true GE-filter are introduced. The characterization of simple very true GE-algebra is discussed, and several properties on very true GE-filter are investigated. Using a very true GE-filter, the quotient very true GE-algebra is constructed, and the uniform and topological space are established.

The notion of K-extending modules was defined recently as a proper generalization of both extending modules and Rickart modules. Let M be a right R-module and let S = EndR(M)...

One of the main goals of science and engineering is to avail human beings cull the maximum propitious decisions. To make these decisions, we need to ken human being’, s predictions, feasible outcomes of various decisions, and since information is never absolutely precise and accurate, we need to withal information about the degree of certainty. All these types of information will lead to partial orders. A partially ordered set (or poset) theory deals with partial orders and plays a major role in real life. It has wide range of applications in various disciplines such as computer science, engineering, medical field, science, modeling spatial relationship in geographic information systems (GIS), physics and so on. In this paper, we mainly focus on weakly primary semi-ideal of a poset. We introduce the concepts of weakly primary semi-ideal and weakly Q-primary semi-ideal for some prime Q of a poset P and characterize weakly primary semi-ideals of P in terms of prime and primary semi-ideals of P. We provide a counter-example for the existence of weakly primary semi-ideal of P which is not a primary semi-ideal of P. We found an equivalent assertion of primary (respy., weakly primary) semi-ideal r(K) for a semi-ideal K of P. Moreover, we introduce the notion of direct product of weakly primary semi-ideal of P and describe its characteristics.

A new sub-structure called (vivid) deductive system is introduced and their properties are examined. Conditions for a subset to be a deductive system are provided. The notion of upper GE-set is also introduced, and an example to show that any upper GE-set may not be a deductive system are supplied. Conditions for an upper GE-set to be a deductive system are provided. An upper GE-set is used to consider conditions for a subset to be a deductive system. The characterization of deductive system is established, and relationship between deductive system and vivid deductive system are created. Conditions for a deductive system to be a vivid deductive system are given, and the extension property for vivid deductive system is constructed.

In this paper we introduce Goldie∗,-δ,-supplemented modules as follows. A module M is called Goldie∗,-δ,-supplemented (briefly, G∗,δ,-supplemented) if there exists a δ,-supplement T of M for every submodule A of M such that Aβ,∗,δ,T. We say that a module M is called Goldie∗,-δ,-lifting (briefly, G∗,δ,-lifting) if there exists a direct summand D of M for every submodule A ofM such that Aβ,∗,δ, D. Note that the last concept given in [4] as a δ,-H-supplemented module. We present fundamental properties of these modules. We indicate that these modules lie between δ,-lifting and δ,-supplemented modules. Also we prove that our modules coincide with some variations of δ,-supplemented modules for δ,-semiperfect modules.

Let R be a commutative Noetherian ring with identity, I be an ideal of R and M be an R-module such that Extj R(R/I, M) is finitely generated for all j...

For a non-abelian group G, its commuting conjugacy class graph CCC(G) is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of G and two distinct vertices xG and yG are adjacent if there exists some elements x ′,∈,xG and y ′,∈,yG such that x ′,y ′,= y ′,x ′, . In this paper we compute the genus of CCC(G) for six well-known classes of non-abelian two-generated groups (viz. D2n, SD8n, Q4m, V8n, U(n, m) and G(p, m, n)) and determine whether CCC(G) for these groups are planar, toroidal, doubletoroidal or triple-toroidal.

In this paper, the authors have represented the genetic structures in terms of automata. With the algebraic structure defined on the genetic code authors defined an automaton on those codons as Σ,= (CG, P, AM, F, G) where P is the set of the four bases A, C, G, U as a set of alphabets or inputs, CG is the set of all 64 codons, obtained from the ordering of the elements of P, as the set of states, AM is the set of the 20 amino acids as the set of outputs that produce during the process. F and G are transition function and output function respectively. Authors observed that M(Σ, ) = ({fa: a ∈,P}, ◦, ) defined on the automata Σ,where fa(q) = F(q, a) = qa, q ∈,CG, a ∈,P is a monoid called the syntactic monoid of Σ, , with fa ◦,fb = fba ∀, a, b ∈,P. Studying the structure defined in terms of automata it is also observed that the algebraic structure (M(CG), +, ·, ) forms a Near-Ring with respect to the two operations ′,+ ′,and ′, ·, ′,where M(CG) = {f|f: CG →,CG}.

The unitary addition Cayley graph Gn[ω, ] of Eisenstein integers modulo n has the vertex set En[ω, ], the set of Eisenstein integers modulo n. Any two vertices x = a1 + ω, b1, y = a2 + ω, b2 of Gn[ω, ] are adjacent if and only if gcd(N(x + y), n) = 1, where N is the norm of any element of En[ω, ] given by N(a + ω, b) = a2 + b2 −,ab. In this paper we obtain some basic graph invariants such as degree of the vertices, number of edges, diameter, girth, clique number and chromatic number of unitary addition Cayley graph of Eisenstein integers modulo n. This paper also focuses on determining the independence number of the above mentioned graph.

Let F be a free group, V be a variety of groups defined by the set of laws V ⊆,F and G be a finite V-nilpotent p-group. The automorphism α,of G is said to be a marginal automorphism (with respect to V ), if for all x ∈,G, x −, 1xα,∈,V ⋆, (G), where V ⋆, (G) denotes the marginal subgroup of G. An automorphism α,of G is called an IA-automorphism if x −, 1xα,∈,G ′,for each x ∈,G. An automorphism α,of G is called a class preserving if for all x ∈,G, there exists an element gx ∈,G such that xα,= g −, 1 x xgx. Let AutV ⋆,(G), AutG ′,(G) and Autc(G) respectively, denote the group of all marginal automorphisms, IA-automorphisms and class preserving automorphisms of G. In this paper, first we give a necessary and sufficient condition on a finite V-nilpotent p-group G such that each marginal automorphism of G fixes the center of G element-wise. Then we characterize all finite V-nilpotent p-groups G such that AutV ⋆,(G) = AutG ′,(G). Finally, we obtain a necessary and sufficient condition for a finite V-nilpotent p-group G such that AutV ⋆,(G) = Autc(G).

The notion of ω,-filters is introduced in distributive lattices and their properties are studied. A set of equivalent conditions is derived for every maximal filter of a distributive lattice to become an ω,-filter which leads to a characterization of quasi-complemented lattices. Some sufficient conditions are derived for proper D-filters of a distributive lattice to become an ω,-filter. Finally, ω,-filters of a distributive lattice are characterized with the help of minimal prime D-filters.