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We prove that a unital completely positive map between finite-dimensional C ⁎ -algebras is a homomorphism if and only if it is completely entropy-nonincreasing, where the relevant notion of entropy is a variant of von Neumann entropy. This adjusted von Neumann entropy is the negative of the relative entropy with respect to the uniform state on the C ⁎ -algebra, up to an additive constant. As an intermediate step, we prove that a unital completely positive map between finite-dimensional C ⁎ -algebras is a homomorphism if and only if its adjusted Choi operator is a projection. Both equivalences generalize familiar facts about stochastic maps between finite sets. [ABSTRACT FROM AUTHOR]

Let F be an algebraically closed field with char (F) = 0. The special linear algebra sl 2 is the F -Lie algebra with Chevalley basis { e , h , f }. Since the special orthogonal algebra so 4 is isomorphic to sl 2 ⊕ sl 2 , so 4 is viewed as the F -Lie algebra with Chevalley basis { e 1 , h 1 , f 1 , e 2 , h 2 , f 2 }. In [21, Lemma 3.1] , there is an automorphism ⁎ : so 4 → so 4 so that { e 1 ⁎ , h 1 ⁎ , f 1 ⁎ , e 2 ⁎ , h 2 ⁎ , f 2 ⁎ } is another Chevalley basis of so 4. In [21, Section 5] , there is a simple construction of a finite-dimensional irreducible so 4 -module V on which so 4 acts by derivation. In this paper, we construct four tridiagonal pairs (or TD pairs) on V via the action of the Chevalley bases of so 4. We prove that these TD pairs are of Krawtchouk type and are not necessarily pairwise isomorphic based on their associated Drinfel'd polynomials. Consequently, we display four Lie algebra homomorphisms from the tetrahedron algebra ⊠ to so 4 and via these homomorphisms, we describe how the generators of ⊠ act on V. Finally, we show that the irreducible so 4 -module V is isomorphic to a tensor product of two evaluation modules in view of [16, Definition 4.11]. [ABSTRACT FROM AUTHOR]

Given a field k and a subgroup H of a finite group G, we define a duality for quotients of homomorphisms of kG-modules modulo kG-homomorphisms that factors through a relatively H-projective kG-module. This generalizes the classical Auslander–Reiten duality for group algebras. As an application, we use the generalized Auslander–Reiten duality to give a new way for constructing almost split sequences. [ABSTRACT FROM AUTHOR]

We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive ∃ ∀ -sentence true in 퐅 3 and false in 퐅 4 . Secondly, we show that every model of Th ⁢ (퐅 n) admits a canonical homomorphism into the profinite-bounded completion 퐇 n of 퐅 n . Thirdly, we show that 퐇 n is isomorphic to the Dedekind–MacNeille completion of 퐅 n , and that 퐇 n is not positively elementarily equivalent to 퐅 n , as there is a positive ∀ ∃ -sentence true in 퐇 n and false in 퐅 n . Finally, we show that DM ⁢ (퐅 n) is a retract of Id ⁢ (퐅 n) and that for any lattice 퐊 which satisfies Whitman's condition (W) and which is generated by join prime elements, the three lattices 퐊 , DM ⁢ (퐊) , and Id ⁢ (퐊) all share the same positive universal first-order theory. [ABSTRACT FROM AUTHOR]

Let R be a ℤ -graded ring and let ∗ (R) be the category of ℤ -graded R -modules and homogeneous homomorphisms. In this paper, we define and study some objects in this category. More precisely, we introduce the concepts of graded duo (weak and strong graded duo) modules and give some sources and an example for these types of modules. It is seen that, under some condition, graded duo property is a local property in this category. When the ring R is a discrete graded valuation ring with unique ∗ maximal ideal , we see that these three types of graded (duo) modules are identical and give an explicit characterization of them, so that any graded duo modules over such a ring is of the form ∗ E (R /) (a) or (R / n) (b) for some positive integer n and some integers a , b. The same task is done whenever R is a graded Dedekind domain. Finally, by an example, that provides a wide variety of strong graded duo modules, it was shown that the given characterizations do not hold valid if the ground ring is not Dedekind. [ABSTRACT FROM AUTHOR]

For |$B \subseteq \mathbb F_q^m$|⁠ , the n th affine extremal number of B is the maximum cardinality of a set |$A \subseteq \mathbb F_q^n$| with no subset, which is affinely isomorphic to B. Furstenberg and Katznelson proved that for any |$B \subseteq \mathbb F_q^m$|⁠ , the n th affine extremal number of B is |$o(q^n)$| as |$n \to \infty$|⁠. By counting affine homomorphisms between subsets of |$\mathbb F_q^n$|⁠ , we derive new bounds and give new proofs of some previously known bounds for certain affine extremal numbers. At the same time, we establish corresponding supersaturation results. We connect these bounds to certain Ramsey-type numbers in vector spaces over finite fields. For |$s,t \geq 1$|⁠ , let |$R_q(s,t)$| denote the minimum n such that in every red–blue coloring of the one-dimensional subspaces of |$\mathbb F_q^n$|⁠ , there is either a red s -dimensional subspace or a blue t -dimensional subspace of |$\mathbb F_q^n$|⁠. The existence of these numbers is a special case of a well-known theorem of Graham, Leeb and Rothschild. We improve the best-known upper bounds on |$R_2(2,t)$|⁠ , |$R_3(2,t)$|⁠ , |$R_2(t,t)$| and |$R_3(t,t)$|⁠. [ABSTRACT FROM AUTHOR]

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group G , where conjugacy classes of the boundary components of the surface must map to prescribed conjugacy classes in G , to a sum over values of irreducible characters of G weighted by Frobenius-Schur multipliers. The proof is structured so that the corresponding results for closed and possibly orientable surfaces, as well as some generalizations, are derived using the same methods. We then apply these results to the specific case of the symmetric group. [ABSTRACT FROM AUTHOR]

Let G be a split connected reductive group defined over $\mathbb {Z}$. Let F and $F'$ be two non-Archimedean m -close local fields, where m is a positive integer. D. Kazhdan gave an isomorphism between the Hecke algebras $\mathrm {Kaz}_m^F :\mathcal {H}\big (G(F),K_F\big) \rightarrow \mathcal {H}\big (G(F'),K_{F'}\big)$ , where $K_F$ and $K_{F'}$ are the m th usual congruence subgroups of $G(F)$ and $G(F')$ , respectively. On the other hand, if $\sigma $ is an automorphism of G of prime order l , then we have Brauer homomorphism $\mathrm {Br}:\mathcal {H}(G(F),U(F))\rightarrow \mathcal {H}(G^\sigma (F),U^\sigma (F))$ , where $U(F)$ and $U^\sigma (F)$ are compact open subgroups of $G(F)$ and $G^\sigma (F),$ respectively. In this article, we study the compatibility between these two maps in the local base change setting. Further, an application of this compatibility is given in the context of linkage – which is the representation theoretic version of Brauer homomorphism. [ABSTRACT FROM AUTHOR]

We consider subalgebras A of the algebra M n of n × n complex matrices that contain all diagonal matrices, known in the literature as the structural matrix algebras (SMAs). Let A ⊆ M n be an arbitrary SMA. We first show that any commuting family of diagonalizable matrices in A can be intrinsically simultaneously diagonalized (i.e. the corresponding similarity can be chosen from A). Using this, we then characterize when one SMA Jordan-embeds into another and in that case we describe the form of such Jordan embeddings. As a consequence, we obtain a description of Jordan automorphisms of SMAs, generalizing Coelho's result on their algebra automorphisms. Next, motivated by the results of Marcus-Moyls and Molnar-Šemrl, connecting the linear rank-one preservers with Jordan embeddings M n → M n and T n → M n (where T n is the algebra of n × n upper-triangular matrices) respectively, we show that any linear unital rank-one preserver A → M n is necessarily a Jordan embedding. As the converse fails in general, we also provide a necessary and sufficient condition for when it does hold true. Finally, we obtain a complete description of linear rank preservers A → M n , as maps of the form X ↦ S (P X + (I − P) X t) T , for some invertible matrices S , T ∈ M n and a central idempotent P ∈ A. [ABSTRACT FROM AUTHOR]

Molodtsov pioneered the notion of soft set theory, presenting it as a mathematical tool for dealing with uncertainty. Numerous researchers have subsequently developed models leveraging this theory to tackle challenges in decision-making and medical diagnosis. Soft set theory emerges as a flexible framework adept at handling uncertain and imprecise information, a domain where classical set theory often struggles. Expanding on the soft set concept, researchers have introduced the idea of a soft graph. This innovative concept allows for the creation of diverse representations of graph-based relations by incorporating parameterisation. In this work, we present and investigate some of the features of the homomorphic and restricted homomorphic products of soft graphs. This paper establishes the structural properties of these products, ensuring that they are well-defined and maintain the essential characteristics of soft graphs. Additionally, we derive combinatorial identities related to the counts of vertices and edges, as well as the degree sums, offering deeper insights into the composition and behaviour of these graph products. [ABSTRACT FROM AUTHOR]

We describe the ordering of a class of clones by minion homomorphisms, also known as minor preserving maps or height 1 clone homomorphisms. The class consists of all clones on finite sets determined by binary relations whose projections to both coordinates have at most two elements. This class can be alternatively described up to minion homomorphisms as the class of multisorted Boolean clones determined by binary relations. We also introduce and apply the concept of a minion core which provides canonical representatives for equivalence classes of clones, more generally minions, on finite sets. [ABSTRACT FROM AUTHOR]

In this paper, a kind of restricted wreath product of an arbitrary band of groups by an arbitrary generalized inverse semigroup is introduced first. This wreath product turns out to be an E-solid semigroup, and if one comes out at the beginning with a normal band of groups, then it shows itself to be also a locally inverse semigroup. Then it is proved that every E-solid locally inverse semigroup S can be embedded in such a restricted wreath product of a normal band of groups Q by a generalized inverse semigroup K. Here K can be taken to be the greatest orthodox semigroup homomorphic image of S and Q can be taken to be the preimage of the band of idempotents of K under the canonical homomorphism of S onto its quotient K. In this way, the given E-solid locally inverse semigroup S decomposes into a restricted wreath product of its components Q and K whose structure derives directly from that of S. [ABSTRACT FROM AUTHOR]

We prove that if f : R → S is a local homomorphism of noetherian local rings, and M is a non-zero finitely generated or artinian S-module whose injective dimension over R is bounded by the difference of the embedding dimensions of R and S, then M is an injective S-module and f is an exceptional complete intersection map. [ABSTRACT FROM AUTHOR]

In this article, we answer two questions of Buchanan–McKean [ KSp-characteristic classes determine Spinh cobordism , https://arxiv.org/ abs/2312.08209, 2023] about bordism for manifolds with \mathrm {spin}^h structures: we establish a Smith isomorphism between the reduced \mathrm {spin}^{h} bordism of \mathbb {RP}^\infty and \mathrm {pin}^{h-} bordism, and we provide a geometric explanation for the isomorphism \Omega _{4k}^{\mathrm {Spin}^c}\otimes \mathbb {Z}[1/2] \cong \Omega _{4k}^{\mathrm {Spin}^h}\otimes \mathbb {Z}[1/2]. Our proofs use the general theory of twisted spin structures and Smith homomorphisms that we developed joint with Devalapurkar, Liu, Pacheco-Tallaj, and Thorngren [ The Smith fiber sequence and invertible field theories , https://arxiv.org/abs/2405.04649, 2024], specifically that the Smith homomorphism participates in a long exact sequence with explicit, computable terms. [ABSTRACT FROM AUTHOR]

We introduce the concepts of generalized compatible and cocompatible bimodules in order to characterize Gorenstein projective, injective and flat modules over trivial ring extensions. Let R ⋉ M be a trivial extension of a ring R by an R - R -bimodule M such that M is a generalized compatible R - R -bimodule and Z (R) is a generalized compatible R ⋉ M - R ⋉ M -bimodule. We prove that (X , α) is a Gorenstein projective left R ⋉ M -module if and only if the sequence M ⊗ R M ⊗ R X → M ⊗ α M ⊗ R X → α X is exact and coker(α) is a Gorenstein projective left R -module. Analogously, we explicitly characterize Gorenstein injective and flat modules over trivial ring extensions. As an application, we describe Gorenstein projective, injective and flat modules over Morita context rings with zero bimodule homomorphisms. [ABSTRACT FROM AUTHOR]

We introduce the concept of complex cubic Q neutrosophic subbisemiring (CCQNSBS) is a new extension of cubic Q neutrosophic subbisemiring. We examine the characteristics and homomorphic features of CCQNSBS. We communicate the CCQNSBS level sets for bisemirings. A cubic complex Q neutrosophic subset Γ of bisemiring S if and only if each non-empty level set R(ℓ,♭), where ... is a CCQNSBS of S. We show that the intersection of all CCQNSBSs yields a CCQNSBS of S. If Θ1, Θ2, ..., Θn be the finite collection of CCQNSBSs of S1, S2, ..., Sn respectively. Then Θ1 x Θ2 x ... x Θn is a CCQNSBS of S1 x S2 x ... x Sn. If F : S1 → S2 is a homomorphism, then ... is a subbisemiring of CCQNSBS ... of S2. Examples are provided to show how our findings are used. [ABSTRACT FROM AUTHOR]

In this paper, we have put forward the SH-homo and 0-preserving SH-homo to compare two SH-groupoids along with few examples. Some properties of SH-homo and 0-preserving SH-homo are explored. Also, we proved the SH-homo between two SuperHyper BCI-Algebra is 0-preserving SH-homo. Finally, the category of SuperHyper Groupoid, SuperHyper BCI-Algebra and Neutrosophic SuperHyper BCI-Algebra were investigated. [ABSTRACT FROM AUTHOR]

The notion of the complex Q bipolar neutrosophic subbisemiring (CQBNSBS) is a fundamental notion to be considered for tackling tricky and intricate information. Here, in this study, we want to expand the notion of CQBNSBS by giving a general algebraic structure for tackling bipolar complex fuzzy (BCF) data by fusing the conception of CQBNSBS and subbisemiring. Keeping in view the importance of fuzzy algebraic structures, in this manuscript, we develop the concept of CQBNSBS.We analyze the important properties and homomorphic aspects of CQBNSBS. For bisemirings, we propose the CQBNSBS level sets. We also develop the notions of homomorphic images of all CQBNSBSs is also CQBNSBS and homomorphic pre-images of all CQBNSBSs is also CQBNSBS. Examples are provided to demonstrate our findings. [ABSTRACT FROM AUTHOR]

In this paper, we have defined the concept of two-fold maximal units in finite two-fold neutrosophic rings modulo integers, where a sufficient and necessary condition for such class of generalized units will be provided. We characterize all maximal units in the following two-fold neutrosophic rings ... for n ∈ {2,3,4,5}. [ABSTRACT FROM AUTHOR]

We construct and analyze the concept of complex cubic anti neutrosophic subbisemiring (ComCANSBS). We analyze the important properties and homomorphic aspects of ComCANSBS. For bisemirings, we propose the ComCANSBS level sets. A complex neutrosophic subset of bisemiring S is represented by the symbol Γ if and only if each non-empty level set R(℘,κ), where ... is a ComCANSBS of S. Let Y be a ComCANSBS of bisemiring S. If and only if Y is a ComCANSBS of S x S, then Γ is a ComCANSBS of bisemiring S. Let Γ be the strongest complex anti neutrosophic relation of bisemiring S. We show that homomorphic images of all ComCANSBSs are ComCANSBSs, and homomorphic pre-images of all ComCANSBSs are ComCANSBSs. There are examples given to illustrate our results. [ABSTRACT FROM AUTHOR]

We propose a new theory based on the notions of marginal and large difference which has natural models in the context of nonstandard mathematics. We introduce the notion of finite marginality and show a representation result which ensures, for finitely marginal countable models, the existence of a homomorphism of the structure of marginal and large difference into a nonstandard model of the natural numbers, and show the extent to which any such homomorphism is unique. Finally, we show that our theory constitutes part of the underlying abstract structure of three distinct philosophical theories of vagueness: Dean's neofeasibilism, Itzhaki's theory of nonstandard heuristics, and our own initial sketch of a nonstandard primitivism about vagueness. [ABSTRACT FROM AUTHOR]

AbstractIn recent works, the concept of proximal groups was introduced as a generalization of topological groups. In this paper, we define the concept of quotient proximal groups with the help of canonical mappings, which is not really the generalization of quotient topological groups, in general. We study some topological properties of this quotient proximal group. We prove some important theorems namely induced mapping theorem, first isomorphism theorem and second isomorphism theorem for proximal groups. We also study numerous applications of first isomorphism theorem. The study is supported by various examples. [ABSTRACT FROM AUTHOR]

Let ℓ ≥ 5 {\ell\geq 5} be a prime number and let 픽 ℓ {\mathbb{F}_{\ell}} denote the finite field with ℓ {\ell} elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to GL 2 ⁡ ( 픽 ℓ ) {\operatorname{GL}_{2}(\mathbb{F}_{\ell})} and absolute discriminant bounded above by X is asymptotically at least X ℓ / ( 12 ⁢ ( ℓ - 1 ) ⁢ # ⁢ GL 2 ⁡ ( 픽 ℓ ) ) log ⁡ X {\frac{X^{{\ell}/({12(\ell-1)\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})})}}{% \log X}} . We also obtain a similar result for the number of surjective homomorphisms ρ : Gal ⁡ ( ℚ ¯ / ℚ ) → GL 2 ⁡ ( 픽 ℓ ) {\rho:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{% GL}_{2}(\mathbb{F}_{\ell})} ordered by the prime to ℓ {\ell} part of the Artin conductor of ρ. [ABSTRACT FROM AUTHOR]

Given a cyclically ordered free Abelian group G, a unital C+- algebra A, and a semigroup homomorphism α from P(G) into the semigroup Endo(A), we characterize faithful representations of crossed product A xα P(G) by endomorphisms. [ABSTRACT FROM AUTHOR]

Let N ≥ 4. We prove that every injective homomorphism from the Torelli subgroup IAN to Out(FN) differs from the inclusion by a conjugation in Out(FN). This applies more generally to the following subgroups of Out(FN): every finite-index subgroup of Out(FN) (recovering a theorem of Farb and Handel); every subgroup of Out(FN) that contains a finite-index subgroup of one of the groups in the Andreadakis-Johnson filtration of Out(FN); every subgroup that contains a power of every linearly-growing automorphism; more generally, every twist-rich, subgroup of Out(FN) - those are subgroups that contain sufficiently many twists in an appropriate sense. Among applications, this recovers the fact that the abstract commensurator of every group above is equal to its relative commensurator in Out(FN); it also implies that all subgroups in the Andreadakis-Johnson filtration of Out(FN) are co-Hopfian. We also prove the same rigidity statement for subgroups of Out(F3) which contain a power of every Nielsen transformation. This shows, in particular, that Out(F3) and all its finite-index subgroups are co-Hopfian, extending a theorem of Farb and Handel to the N = 3 case. [ABSTRACT FROM AUTHOR]

Let M n be the algebra of n × n complex matrices. We consider arbitrary subalgebras A of M n which contain the algebra of all upper-triangular matrices (i.e. block upper-triangular subalgebras), and their Jordan embeddings. We first describe Jordan embeddings ϕ : A → M n as maps of the form ϕ (X) = T X T − 1 or ϕ (X) = T X t T − 1 , where T ∈ M n is an invertible matrix, and then we obtain a simple criteria of when one block upper-triangular subalgebra Jordan-embeds into another (and in that case we describe the form of such embeddings). As a main result, we characterize Jordan embeddings ϕ : A → M n (when n ≥ 3) as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless A = M n when injectivity is superfluous). [ABSTRACT FROM AUTHOR]

An element x of an ordered semigroup (S, ·, ≤) is called an inverse element of α ∈ S if α ≤ xαx and x ≤ xαx. An inverse ordered semigroup is an ordered semigroup S for which every element of S possesses an inverse element and the inverse elements of any element of S are in the same connected component of the Hasse diagram of its order relation. We present properties satisfied by inverse ordered semigroups and characterize inverse ordered semigroups based on the notion of regularity of ordered semigroups, their set of idempotents and Green’s relations. Also we prove that the inversibility of ordered semigroups is preserved by homomorphisms. [ABSTRACT FROM AUTHOR]

Given a Noetherian ring A, the collection of integrally closed ideals in A which contain a nonzerodivisor forms a cancellative monoid under the operation I * J = IJ ¯ , the integral closure of the product. The monoid is torsion-free and atomic. Restricting to monomial ideals in a polynomial ring, there is a surjective homomorphism from the Integral Polytope Group onto the Grothendieck group of integrally closed monomial ideals under translation invariance of their Newton Polyhedra. The Integral Polytope Group, the Grothendieck group of polytopes with integer vertices under Minkowski addition and translation invariance, has an explicit basis, allowing for explicit factoring in the monoid. [ABSTRACT FROM AUTHOR]

For a commutative semiring S, by an S-algebra we mean a commutative semiring A equipped with a homomorphism S → A. We show that the subvariety of S-algebras determined by the identities 1 + 2x = 1 and x² = x is closed under non-empty colimits. The (known) closedness of the category of Boolean rings and of the category of distributive lattices under non-empty colimits in the category of commutative semirings both follow from this general statement. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish that the underlying object functor from the category of models of a Lawvere theory to the base category creates limits which exist in the base category and creates coequalisers of all parallel pairs of homomorphisms whose underlying pairs admit a split coequaliser in the base category. Furthermore, we show that for a small-complete category with a well-behaved proper factorisation structure, the underlying object functor admits a left adjoint, and hence this underlying object functor is monadic in the sense of Beck’s theorem. In particular, this establishes the existence of free localic algebras for any Lawvere theory, generalising the known results for the existence of free localic groups. [ABSTRACT FROM AUTHOR]

We exhibit the proximity frames and proximity homomorphisms as a Kleisli category of a comonad whose underlying functor takes a proximity frame to its frame of round ideals. This construction is known in the literature as stable compactification ([6]). We show that the frame of round ideals naturally carries with it two proximities of interest from which two comonads are induced. [ABSTRACT FROM AUTHOR]

Let N ≥ 4. We prove that every injective homomorphism from the Torelli subgroup IAN to Out(FN) differs from the inclusion by a conjugation in Out(FN). This applies more generally to the following subgroups of Out(FN): every finite-index subgroup of Out(FN) (recovering a theorem of Farb and Handel); every subgroup of Out(FN) that contains a finite-index subgroup of one of the groups in the Andreadakis-Johnson filtration of Out(FN); every subgroup that contains a power of every linearly-growing automorphism; more generally, every twist-rich, subgroup of Out(FN) - those are subgroups that contain sufficiently many twists in an appropriate sense. Among applications, this recovers the fact that the abstract commensurator of every group above is equal to its relative commensurator in Out(FN); it also implies that all subgroups in the Andreadakis-Johnson filtration of Out(FN) are co-Hopfian. We also prove the same rigidity statement for subgroups of Out(F3) which contain a power of every Nielsen transformation. This shows, in particular, that Out(F3) and all its finite-index subgroups are co-Hopfian, extending a theorem of Farb and Handel to the N = 3 case. [ABSTRACT FROM AUTHOR]

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We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker's layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi's regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. However, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular, we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree. [ABSTRACT FROM AUTHOR]

A typical question addressed in this paper is the following. Suppose Z ⊂ Y ⊂ X are hyperbolic spaces where Z is quasiconvex in both Y and X. Let Y ^ and X ^ denote the spaces obtained from Y and X respectively by coning off Z as defined by Farb [16]. If the inclusion of the coned-off spaces Y ^ → X ^ admits the Cannon–Thurston (CT) map then does the inclusion Y → X also admit the Cannon–Thurston map? The main result of this paper answers this question affirmatively provided Y ^ → X ^ satisfies Mitra's criterion [36] (see Lemma 2.32) for the existence of CT maps, although the answer in general is negative. The main application of our theorem is in the context of acylindrical complexes of hyperbolic groups. In Martin [33], he proved a combination theorem for developable, acylindrical complexes of hyperbolic groups. Suppose (G , Y) is an acylindrical complex of hyperbolic groups with universal cover B which satisfy the hypotheses of Martin's theorem. Suppose Y 1 ⊂ Y is a connected subcomplex such that the subcomplex of groups (G , Y 1) also satisfies the hypotheses of Martin's theorem, it has universal cover B 1 and the natural homomorphism π 1 (G , Y 1) → π 1 (G , Y) is injective. It follows from the main theorem of this paper that the inclusion π 1 (G , Y 1) → π 1 (G , Y) admits the CT map if the inclusion B 1 → B satisfies Mitra's criterion. Also π 1 (G , Y 1) is quasiconvex in π 1 (G , Y) if in addition B 1 is qi-embedded in B. [ABSTRACT FROM AUTHOR]

For two given hypergraphs, we construct an augmentation hypergraph. We establish an algebra homomorphism from the Leavitt path algebra of the augmentation hypergraph to the tensor product of the corresponding Leavitt path algebras of the two given hypergraphs. And we apply the homomorphism to certain weighted Leavitt path algebras of weighted graphs. We find a sufficient condition on weighted graphs such that one realizes the weighted Leavitt path algebras of the augmentation for weighted graphs as the tensor product of the weighted Leavitt path algebras. [ABSTRACT FROM AUTHOR]

The notion of graph covers is a discretization of covering spaces introduced and deeply studied in topology. In discrete mathematics and theoretical computer science, they have attained a lot of attention from both the structural and complexity perspectives. Nonetheless, disconnected graphs were usually omitted from the considerations with the explanation that it is sufficient to understand coverings of the connected components of the target graph by components of the source one. However, different (but equivalent) versions of the definition of covers of connected graphs generalize to non-equivalent definitions for disconnected graphs. The aim of this paper is to summarize this issue and to compare three different approaches to covers of disconnected graphs: (1) locally bijective homomorphisms, (2) globally surjective locally bijective homomorphisms (which we call surjective covers), and (3) locally bijective homomorphisms which cover every vertex the same number of times (which we call equitable covers). The standpoint of our comparison is the complexity of deciding if an input graph covers a fixed target graph. We show that both surjective and equitable covers satisfy what certainly is a natural and welcome property: covering a disconnected graph is polynomial-time decidable if such it is for every connected component of the graph, and it is NP-complete if it is NP-complete for at least one of its components. We further argue that the third variant, equitable covers, is the most natural one, namely when considering covers of colored graphs. Moreover, the complexity of surjective and equitable covers differ from the fixed parameter complexity point of view. In line with the current trends in topological graph theory, as well as its applications in mathematical physics, we consider graphs in a very general sense: our graphs may contain loops, multiple edges and also semi-edges. Moreover, both vertices and edges may be colored, in which case the covering projection must respect the colors. We conclude the paper by a complete characterization of the complexity of covering 2-vertex colored graphs, and show that poly-time/NP-completeness dichotomy holds true for this case. We actually aim for a stronger dichotomy. All our polynomial-time algorithms work for arbitrary input graphs, while the NP-completeness theorems hold true even in the case of simple input graphs. [ABSTRACT FROM AUTHOR]

Neutrosophic exact sequence and its construction is presented as algebraic structure in this article. A mathematical construction known as a neutrosophic exact sequence expands the implications of an exact sequence from classical algebra to the domain of neutrosophic sets. This work introduces a full description of neutrosophic exact sequence of R-modules as a structural preserving tools and discusses the fundamental features. Additionally we examine the role of beta level sets in neutrosophic sets and investigate the neutrosophic split exact sequence. [ABSTRACT FROM AUTHOR]

This study discusses the use of t-neutrosophic graphs to express and understand complex interactions. It also shows how these graphs might manage intricate relationships with a wide range of variables or scenario elements. Additionally, it offers fundamental t-neutrosophic graph operations and talks about concepts in these graphs such as homomorphism and isomorphism. Additionally, the study addresses how this approach may be applied in practical contexts to address circular economies and sustainability growth. Through the use of t-neutrosophic graphs, various components, and their connections, the research demonstrates that managing the several circular dimensions is feasible. financial system. This example shows how adaptable and practical t-neutrosophic graphs are as a tool for developing policies and making decisions about complicated social issues that involve some numbers principles. [ABSTRACT FROM AUTHOR]

An (n , m) -graph is characterized by n types of arcs and m types of edges. A homomorphism of an (n , m) -graph G to an (n , m) -graph H , is a vertex mapping that preserves adjacency, direction, and type. The (n , m) -chromatic number of G , denoted by χ n , m (G) , is the minimum value of | V (H) | such that there exists a homomorphism of G to H. The theory of homomorphisms of (n , m) -graphs have connections with graph theoretic concepts like harmonious coloring, nowhere-zero flows; with other mathematical topics like binary predicate logic, Coxeter groups; and has application to the Query Evaluation Problem (QEP) in graph database. In this article, we show that the arboricity of G is bounded by a function of χ n , m (G) but not the other way around. Additionally, we show that the acyclic chromatic number of G is bounded by a function of χ n , m (G) , a result already known in the reverse direction. Furthermore, we prove that the (n , m) -chromatic number for the family of graphs with maximum average degree less than 2 + 2 4 (2 n + m) − 1 , including the subfamily of planar graphs with girth at least 8 (2 n + m) , equals 2 (2 n + m) + 1. This improves upon previous findings, which proved the (n , m) -chromatic number for planar graphs with girth at least 10 (2 n + m) − 4 is 2 (2 n + m) + 1. It is established that the (n , m) -chromatic number for the family T 2 of partial 2-trees is both bounded below and above by quadratic functions of (2 n + m) , with the lower bound being tight when (2 n + m) = 2. We prove 14 ≤ χ (0 , 3) (T 2) ≤ 15 and 14 ≤ χ (1 , 1) (T 2) ≤ 21 which improves both known lower bounds and the former upper bound. Moreover, for the latter upper bound, to the best of our knowledge we provide the first theoretical proof. [ABSTRACT FROM AUTHOR]

A group 퐺 is twisted conjugacy separable if, for every automorphism 휑, distinct 휑-twisted conjugacy classes can be separated in a finite quotient. Likewise, 퐺 is completely twisted conjugacy separable if, for any group 퐻 and any two homomorphisms φ , ψ \varphi,\psi from 퐻 to 퐺, distinct ( φ , ψ ) (\varphi,\psi) -twisted conjugacy classes can be separated in a finite quotient. We study how these properties behave with respect to taking subgroups, quotients and finite extensions, and compare them to other notions of separability in groups. Finally, we show that, for polycyclic-by-nilpotent-by-finite groups, being completely twisted conjugacy separable is equivalent to all quotients being residually finite. [ABSTRACT FROM AUTHOR]

Given a regular covering map φ : Λ → Γ \varphi\colon\Lambda\to\Gamma of graphs, we investigate the subgroup LAut ⁡ ( φ ) \operatorname{LAut}(\varphi) of the automorphism group Aut ⁡ ( A Γ ) \operatorname{Aut}(A_{\Gamma}) of the right-angled Artin group A Γ A_{\Gamma} . This subgroup comprises all automorphisms that can be lifted to automorphisms of A Λ A_{\Lambda} . We first show that LAut ⁡ ( φ ) \operatorname{LAut}(\varphi) is generated by a finite subset of Laurence’s elementary automorphisms. For the subgroup FAut ⁡ ( φ ) \operatorname{FAut}(\varphi) of Aut ⁡ ( A Λ ) \operatorname{Aut}(A_{\Lambda}) that consists of lifts of automorphisms in LAut ⁡ ( φ ) \operatorname{LAut}(\varphi) , there exists a natural homomorphism FAut ⁡ ( φ ) → LAut ⁡ ( φ ) \operatorname{FAut}(\varphi)\to\operatorname{LAut}(\varphi) induced by 휑. We then show that the kernel of this homomorphism is virtually a subgroup of the Torelli subgroup IA ⁡ ( A Λ ) \operatorname{IA}(A_{\Lambda}) and deduce a short exact sequence reminiscent of results from the Birman–Hilden theory for surfaces. [ABSTRACT FROM AUTHOR]

Abstract–In this paper, we study additive maps f:A→A on an alternative Cayley-Dickson algebra A satisfying the product preserving property: f(x)f(y)=m whenever xy = k, where k and m are constant elements of A . We prove that f may be written as a multiple of a Jordan homomorphism. Unlike in the associative case, Jordan homomorphisms of strictly alternative division rings need not be homomorphisms or anti-homomorphisms, but we provide necessary and sufficient conditions for this to occur. We then extend these ideas to split alternative Cayley-Dickson algebras of characteristic not 2 and present some open questions when k or m is noninvertible. The final section handles the problem k=m=0 and generalizes to biadditive maps that preserve the zero product. [ABSTRACT FROM AUTHOR]

By considering a particular type of invariant Seifert surfaces we define a homomorphism Φ$\Phi$ from the (topological) equivariant concordance group of directed strongly invertible knots C∼$\widetilde{\mathcal {C}}$ to a new equivariant algebraic concordance group G∼Z$\widetilde{\mathcal {G}}^\mathbb {Z}$. We prove that Φ$\Phi$ lifts both Miller and Powell's equivariant algebraic concordance homomorphism (J. Lond. Math. Soc. (2023), no. 107, 2025‐2053) and Alfieri and Boyle's equivariant signature (Michigan Math. J. 1 (2023), no. 1, 1–17). Moreover, we provide a partial result on the isomorphism type of G∼Z$\widetilde{\mathcal {G}}^\mathbb {Z}$ and obtain a new obstruction to equivariant sliceness, which can be viewed as an equivariant Fox–Milnor condition. We define new equivariant signatures and using these we obtain novel lower bounds on the equivariant slice genus. Finally, we show that Φ$\Phi$ can obstruct equivariant sliceness for knots with Alexander polynomial one. [ABSTRACT FROM AUTHOR]

In the previous paper, we defined a new category which categorifies the Hecke algebra. This is a generalization of the theory of Soergel bimodules. To prove theorems, the existences of certain homomorphisms between Bott–Samelson bimodules are assumed. In this paper, we prove this assumption. We only assume the vanishing of certain two-colored quantum binomial coefficients. [ABSTRACT FROM AUTHOR]

For every |$m\geq 2$| we produce an example of a non-hyperbolic finitely presented subgroup |$H < G$| of a hyperbolic group |$G$|⁠ , which is the kernel of a surjective homomorphism |$\phi : G\to{\mathbb Z}^{m}$|⁠. The examples we produce are of finiteness type |$F_{2}$| and not |$F_{3}$|⁠. [ABSTRACT FROM AUTHOR]

With the goal of transferring dg algebra structures on complexes along contractions, we introduce a new condition on the associated homotopy, namely a generalized version of the Leibniz rule. We prove that, with this condition, the transfer works to yield a dg algebra (with vanishing descended higher A ∞ products) and prove that it works also after an application of the Perturbation Lemma even though the new homotopy may no longer satisfy that condition. We also extend these results to the setting of A ∞ algebras. Then we return to our original motivation from commutative algebra. We apply these methods to find a new method for building a dg algebra structure on a well-known resolution, obtaining one that is both concrete and permutation invariant. The naturality of the construction enables us to find dg algebra homomorphisms between these as well, enabling them to be used as inputs for constructing bar resolutions. [ABSTRACT FROM AUTHOR]

The skew generalized power series ring R [ [ S , ω ] ] is a ring construction involving a ring R, a strictly ordered monoid (S , ≤) , and a monoid homomorphism ω : S → E n d (R) . The ring R [ [ S , ω ] ] is a common generalization of ring extensions such as (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, (skew) Mal'cev–Neumann series rings, and (skew) monoid rings. In this paper, we study the nilpotent elements of skew generalized power series rings and the relationships between the properties of the rings R and R [ [ S , ω ] ] expressed in terms of annihilators, such as weak symmetry, weak zip, and the nil-Armendariz and McCoy properties. We obtain results on transferring the weak symmetry and weak zip properties between the rings R and R [ [ S , ω ] ] , as well as sufficient and necessary conditions for a ring R to be (S , ω) -nil-Armendariz or linearly (S , ω) -McCoy. [ABSTRACT FROM AUTHOR]

Fully homomorphic encryption (FHE) is capable of handling sensitive encrypted data in untrusted computing environments. The efficient application of FHE schemes in secure outsourced computation can effectively address security and privacy concerns. This paper presents a novel fully homomorphic encryption scheme called GMS, based on the n-secret learning with errors (LWE) assumption. By utilizing block matrix and decomposition technology, GMS achieves shorter encryption and decryption times and smaller ciphertext sizes compared to existing FHE schemes. For secure outsourced matrix multiplication A m × n · B n × l with arbitrary dimensions, GMS only requires O (max { m , n , l }) rotations and one homomorphic multiplication. Compared to the state-of-the-art methods, our approach stands out by achieving a significant reduction in the number of rotations by a factor of O (log max { n , l }) , along with a decrease in the number of homomorphic multiplications by a factor of n and O (min { m , n , l }) . The experimental results demonstrate that GMS shows superior performance for secure outsourced matrix multiplication of any dimension. For example, when encrypting a 64 × 64 -dimensional matrix, the size of the ciphertext is only 1.27 MB. The encryption and decryption process takes approximately 0.2 s. For matrix multiplication A 64 × 64 · B 64 × 64 , the runtime of our method is 39.98 s, achieving a speedup of up to 5X and 2X. [ABSTRACT FROM AUTHOR]