Homomorphic Image Research Articles

Approximation properties of torsion classes

AbstractWe strengthen a result of Bagaria and Magidor (Trans. Amer. Math. Soc. 366 (2014), no. 4, 1857–1877) about the relationship between large cardinals and torsion classes of abelian groups, and prove that the Maximum Deconstructibility principle introduced in Cox (J. Pure Appl. Algebra 226 (2022), no. 5) requires large cardinals; it sits, implication‐wise, between Vopěnka's Principle and the existence of an ‐strongly compact cardinal. While deconstructibility of a class of modules always implies the precovering property by Saorín and Šťovíček (Adv. Math. 228 (2011), no. 2, 968–1007), the concepts are (consistently) nonequivalent, even for classes of abelian groups closed under extensions, homomorphic images, and colimits.

Regular semigroups weakly generated by set

AbstractWe prove the existence and uniqueness (up to isomorphism) of a regular semigroup $${{\,\textrm{F}\,}}(X)$$ F ( X ) weakly generated by a set X of elements such that all other regular semigroups weakly generated by X are homomorphic images of $${{\,\textrm{F}\,}}(X)$$ F ( X ) . The semigroup $${{\,\textrm{F}\,}}(X)$$ F ( X ) is introduced by a presentation and the word problem for that presentation is solved. The structure of the semigroup $${{\,\textrm{F}\,}}(X)$$ F ( X ) is also studied.

Product-complete tilting complexes and Cohen–Macaulay hearts

We show that the cotilting heart associated to a tilting complex T is a locally coherent and locally coperfect Grothendieck category (i.e., an Ind-completion of a small artinian abelian category) if and only if T is product-complete. We then apply this to the specific setting of the derived category of a commutative noetherian ring R . If \dim(R)<\infty , we show that there is a derived duality \mathcal{D}^{\textup{b}}_{\textup{fg}}(R) \cong \mathcal{D}^{\textup{b}}(\mathcal{B})^{\textup{op}} between \mod R and a noetherian abelian category \mathcal{B} if and only if R is a homomorphic image of a Cohen–Macaulay ring. Along the way, we obtain new insights about t-structures in \mathcal{D}^{\textup{b}}_{\textup{fg}}(R) . In the final part, we apply our results to obtain a new characterization of the class of those finite-dimensional noetherian rings that admit a Gorenstein complex.

A note on conjugacy of supplements in soluble periodic linear groups

Abstract The aim of this short note is to prove that if G is a (homomorphic images of a) soluble periodic linear group and N is a locally nilpotent normal subgroup of G such that N and G / N {G/N} have no isomorphic G-chief factors, then two supplements to N in G are conjugate provided that they have the same intersection with N. This result follows from well-known theorems in the theory of Schunck classes (see [A. Ballester-Bolinches and L. M. Ezquerro, On conjugacy of supplements of normal subgroups of finite groups, Bull. Aust. Math. Soc. 89 2014, 2, 293–299]), and it appeared as the main theorem of [C. Parker and P. Rowley, A note on conjugacy of supplements in finite soluble groups, Bull. Lond. Math. Soc. 42 2010, 3, 417–419].

The degree of commutativity of wreath products with infinite cyclic top group

The degree of commutativity of a finite group is the probability that two uniformly and randomly chosen elements commute. This notion extends naturally to finitely generated groups G: the degree of commutativity dcS(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{dc}\\,}}_S(G)$$\\end{document}, with respect to a given finite generating set S, results from considering the fractions of commuting pairs of elements in increasing balls around 1G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1_G$$\\end{document} in the Cayley graph . We focus on restricted wreath products of the form G=H≀⟨t⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G = H \\hspace{1.111pt}{\\wr }\\hspace{1.111pt}\\langle \\hspace{1.111pt}t \\rangle $$\\end{document}, where H≠1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H \ e 1$$\\end{document} is finitely generated and the top group ⟨t⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\hspace{1.111pt}t \\rangle $$\\end{document} is infinite cyclic. In accordance with a more general conjecture, we show that dcS(G)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{dc}\\,}}_S(G) = 0$$\\end{document} for such groups G, regardless of the choice of S. This extends results of Cox who considered lamplighter groups with respect to certain kinds of generating sets. We also derive a generalisation of Cox’s main auxiliary result: in ‘reasonably large’ homomorphic images of wreath products G as above, the image of the base group has density zero, with respect to certain types of generating sets.

A Research on the Generalizations of Modules Whose Submodules are Isomorphic to a Direct Summand

A module M is called virtually semisimple (resp. virtually extending) if every submodule (resp. complement submodule) of M is isomorphic to a direct summand of M. It is known that virtually extending modules is a generalization of virtually semisimple modules. In this paper, the relationships between virtually extending modules and other generalizations of virtually semisimple modules are examined. Moreover, we introduce a new generalization of virtually semisimple modules; namely CH modules: We say a module M is a c-epi-retractable (or briefly CH module) if any complement submodule of M is a homomorphic image of M. CH modules contains the class of virtually extending modules and the class of epi-retractable modules. We also give some basic properties of this new module class.

Binary relations applied to the fuzzy substructures of quantales under rough environment

Abstract Binary relations (BIRs) have many applications in computer science, graph theory, and rough set theory. This study discusses the combination of BIRs, fuzzy substructures of quantale, and rough fuzzy sets. Approximation of fuzzy subsets of quantale with the help of BIRs is introduced. In quantale, compatible and complete relations in terms of aftersets and foresets with the help of BIRs are defined. Furthermore, we use compatible and complete relations to approximate fuzzy substructures of quantale, and these approximations are interpreted by aftersets and foresets. This concept generalizes the concept of rough fuzzy quantale. Finally, using BIRs, quantale homomorphism is used to build a relationship between the approximations of fuzzy substructures of quantale and the approximations of their homomorphic images.

TILTING COMPLEXES AND CODIMENSION FUNCTIONS OVER COMMUTATIVE NOETHERIAN RINGS

AbstractIn the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the “slice” condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen–Macaulay ring. We also provide dual versions of our results in the cosilting case.

On the power of pushing or stationary moves for input-driven pushdown automata

Input-driven pushdown automata (IDPDAs) are pushdown automata where the next action on the pushdown store (push, pop, nothing) is solely governed by the input symbol. Nowadays such devices are usually defined such that every push operation pushes exactly one additional symbol on the pushdown store and, in addition, stationary moves are not allowed so that the devices work in real time. Here, we relax this strong definition and consider IDPDAs that may push more than one symbol in one step (push-IDPDA) or may perform stationary moves (stat-IDPDA). We study the computational power of the extended variants both in the deterministic and nondeterministic case, we investigate several decidability questions for the new automata classes, and we obtain interesting representations by inverse homomorphisms. Namely, every (1) deterministic, (2) real-time deterministic, and (3) nondeterministic context-free language can be characterized as the inverse homomorphic image of a language accepted by a (1) stat-IDPDA, (2) push-IDPDA, and (3) nondeterministic push-IDPDA.

Congruence filter pairs, equational filter pairs and adjoints

Abstract Filter pairs are a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, given by presenting its lattice of theories as the image of a lattice homomorphism, with certain properties ensuring that the resulting logic is substitution invariant. Every substitution invariant logic arises from a filter pair. Particular classes of logics can be characterized as arising from special classes of filter pairs. We consider so-called congruence filter pairs, i.e. filter pairs for which the domain of the lattice homomorphism is a lattice of congruences for some quasivariety. We show that the class of logics admitting a presentation by such a filter pair is exactly the class of logics having an algebraic semantics. We study the properties of a certain Galois connection coming with such filter pairs. We give criteria for a congruence filter pair to present a logic in some classes of the Leibniz hierarchy by means of this Galois connection, and its interplay with the Leibniz operator.

Not all nilpotent monoids are finitely related

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely related. To our knowledge, this is the first example of a finitely related semigroup where adjoining an identity yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related.

Hybrid bi-ideals in Near- Subtraction Semigroups

The fuzzy set is an excellent solution for dealing with ambiguity and for expressing people’s hesitation in regular life. Soft set theory is an innovative method for solving practical issues. This is useful in resolving a number of problems, and a lot of progress is being made at the moment. In order to develop hybrid structures, Jun et al. fused the fuzzy and soft sets. In this paper, the notion of hybrid bi-ideals in near-subtraction semigroups is proposed and their associated results are discussed. The notion of hybrid intersections is examined. Furthermore, we establish some results related to the homomorphic image and preimage of a hybrid bi-ideal in near-subtraction semigroups.

Amply socle supplemented modules

In this work, amply socle supplemented modules are defined and some properties of these modules are investigated. We prove that every π− projective and socle supplemented module is amply socle supplemented. We also prove that every factor module and every homomorphic image of an amply socle supplemented module are amply socle supplemented. Let M be a projective and socle supplemented R−module. Then every finitely M−generated R−module is amply socle supplemented.

Extending the concepts of complex interval valued neutrosophic subbisemiring of bisemiring

The objective of this paper is to investigate the innovative concept of complex neutrosophic subbisemiring. The novelty of the complex neutrosophic subbisemiring lies in its wide range of truth, indeterminacy, and false function values. It goes beyond the range of [0,1] in the complex plane in contrast to the traditional range [0,1]. Therefore, these three functions can be described mathematically using a complex number in the complex neutrosophic subbisemiring. We develop and analyze the concept of complex interval-valued neutrosophic subbisemiring (CIVNSBS). Moreover, we study homomorphic characteristics and important properties of CIVNSBS. We propose the level sets of CIVNSBS and complex interval valued neutrosophic normal subbisemiring (CIVNNSBS) of bisemirings. Moreover, we introduce CIVNNSBS of bisemiring. Let ¡ be a complex neutrosophic subset of bisemiring S. Then is a CIVNSBS of S if and only if all non empty level set is a subbisemiring, where . Let ¡be a CIVNSBS of bisemiring S and V be the strongest complex neutrosophic relation of bisemiring S. Then ¡ is a CIVNSBS of bisemiring S if and only if V is a CIVNSBS of . We illustrate that homomorphic images of every CIVNSBS is a CIVNSBS and homomorphic pre-images of every CIVNSBS is a CIVNSBS. Examples are provided to illustrate our results.\\

New algebraic approach towards interval-valued neutrosophic cubic vague set based on subbisemiring over bisemiring

We introduces the concept of a interval-valued neutrosophic cubic vague subbisemiring (IVNCVSBS), level sets of IVNCVSBS of a bisemiring. IVNCVSBSs are the new extension of neutrosophic subbisemirings and SBS over bisemirings. Let be a neutrosophic vague subset in $X$, we show that is a IVNCVSBS of X if and only if all non empty level set is a SBS of X. Let be a IVNCVSBS of a bisemiring X and strongest cubic neutrosophic vague relation of X, we prove that is a IVNCVSBS of X × X. Let be any IVNCVSBS of X, prove that pseudo cubic neutrosophic vague coset is a IVNCVSBS of X. Let 1, 2,...,n be the family of IVNCVSBS of X1, X2,..., Xn respectively.The homomorphic image of every IVNCVSBS is a IVNCVSBS. The homomorphic pre-image of every IVNCVSBS is a IVNCVSBS. Examples are provided to strengthen our results.

New approach to bisemiring via the q-neutrosophic cubic vague subbisemiring

We introduce the notion of q-neutrosophic cubic vague subbisemiring (q-NSCVSBS) and level set of q- NSCVSBS of a bisemiring. The q-NSCVSBS is a new concept of subbisemirings of bisemirings. Let X be a neutrosophic vague subset of L. Then W = ([T-, T+ ], [I-, I+ ],[F-,F+ ]) is a q-NSCVSBS of L if and only if all non-empty level set is also a SBS of L. Let X be the q-NSCVSBS of L and ¡ be the strongest cubic q-neutrosophic vague relation of L*L. Then X is a q-NSCVSBS of L* L. Let X be the q-NSCVSBS of L, show that pseudo cubic q-neutrosophic vague coset is also a q-NSCVSBS of L. Let X1, X2,….. Xn be the any family of q-NSCV SBSs of L1, L2,…., Ln respectively, then X1* X2 *….. * Xn is also a q-NSCVSBS of L1 * L2 *…. *Ln .The homomorphic image of every q-NSCVSBS is also a q-NSCVSBS. The homomorphic pre-image of every q-NSCVSBS is also a q-NSCVSBS.

Abelian subgroups based on neutrosophic sets

The notion of a neutrosophic Abelian subgroup of a group is introduced. The characterizations of a neutrosophic Abelian subgroup are investigated. We show that the homomorphic preimage of a neutrosophic Abelian subgroup of a group is a neutrosophic Abelian subgroup, and the onto homomorphic image of a neutrosophic Abelian subgroup of a group is a neutrosophic Abelian subgroup.

Interval Valued Neutrosophic Subalgebra in INK-Algebra

This work presents the concept of interval-valued neutrosophic INK-subalgebras, also known as IV N INKsubalg’s, which are the level and strong level neutrosophic INK-subalgebras. Next, we establish and validate a few theorems that establish the connection between these concepts and neutrosophic INK-subalgebras. We define the images and inverse images of IV N INK-subalgebras and study the transformations of the homomorphic images and inverse images of interval valud neutrosophic (briefly, IV N) INK-subalgebra into IV N INK-subalgebras.

Efficient and privacy-preserving image classification using homomorphic encryption and chunk-based convolutional neural network

Image feature categorization has emerged as a crucial component in many domains, including computer vision, machine learning, and biometrics, in the dynamic environment of big data and cloud computing. It is extremely difficult to guarantee image data security, privacy, and computing efficiency while also lowering storage and transmission costs. This paper introduces a novel method for classifying image features that combines multilevel homomorphic encryption and image data partitioning in an integrated manner. We employ a novel partitioning strategy to reduce computational complexity, significantly reducing computational load and improving classification accuracy. In the quest for increased data security and privacy, we introduce a novel, fully homomorphic encryption approach specialized to partitioned images. To counter the inherent complexity of encryption, we devise a compound encryption strategy that exploits the full potential of homomorphic computation, with an explicit objective to curtail computational and storage overheads. Evidently superior to conventional methods, our methodology showcases pronounced benefits in computational efficiency, storage and transmission cost reduction, and robust security and privacy preservation. Hence, the methodology put forth in this paper presents a pioneering and efficacious resolution to the multifaceted challenges of image feature classification within the intricate milieu of cloud computing and big data.

On Inverse Protocols of Post Quantum Cryptography Based on Pairs of Noncommutative Multivariate Platforms Used in Tandem


 
 
 Non-commutative cryptography studies cryptographic primitives and systems which are based on algebraic structures like groups, semigroups and noncommutative rings. We continue to investigate inverse protocols of Non-commutative cryptography defined in terms of subsemigroups of Affine Cremona Semigroups over finite fields or arithmetic rings Zm and homomorphic images of these semigroups as possible instruments of Post Quantum Cryptography. This approach allows to construct cryptosystem which are not public keys, when protocol finish correspondents have mutually inverse transformations on affine space Kn or variety (K*)n where K is the field or arithmetic ring.
 The security of such inverse protocol rests on the complexity of word problem to decompose element of Affine Cremona Semigroup given in its standard form into composition of given generators. We discuss the idea of usage combinations of two cryptosystems with cipherspaces(K*)n and Kn to form a new cryptosystem with the plainspace(K*)n, ciphertextKn and nonbijective highly nonlinear encryption map.