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An even Artin group is a group which has a presentation with relations of the form ( s t ) n = ( t s ) n with n ≥ 1 . With a group G we associate a Lie Z -algebra TG r ( G ) . This is the usual Lie algebra defined from the lower central series, truncated at the third rank. For each even Artin group G we determine a presentation for TG r ( G ) . By means of this presentation we obtain information about the diagram of G. We then prove an isomorphism criterion for Coxeter matrices that ensures that the diagram of G is uniquely determined by this information. Let d ≥ 2 . We show that, if two even Artin groups G and H having presentations with relations of the form ( s t ) d k = ( t s ) d k with k ≥ 0 are such that TG r ( G ) ≃ TG r ( H ) , then G and H have the same presentation up to permutation of the generators. On the other hand, we show an example of two non-isomorphic even Artin groups G and H such that TG r ( G ) ≃ TG r ( H ) .

The main purpose of this paper is to show that injective balanced L- S 0 -convex spaces and fuzzy frames are isomorphic from the categorical point of view. Meanwhile, we get that every fuzzy frame equipped with the strong L-filter convex structure is an injective balanced L- S 0 -convex space and conversely, the specialization L-ordered set of an injective balanced L- S 0 -convex space is a fuzzy frame.

This paper establishes a theory of calculus for fuzzy-number-valued functions whose range is embedded in a specific Banach subspace of fuzzy numbers. These spaces are generated by strongly linearly independent (SLI) fuzzy numbers whose operations are induced by an isomorphism with R n . We use the notion of Frechet differentiability and Riemann integrability for these functions and present a fundamental theorem of calculus. Lastly, we develop a theory of fuzzy differential equations and provide an existence and uniqueness theorem.

The Rueda's algebras R ( f , ς ) (simply R) are a class of algebras similar to the universal enveloping algebra of sl 2 . We study the category H of R-modules whose objects are free of rank 1 when restricted to R 0 = C [ H ] . We classify the isomorphism classes of objects in H and determine the simplicity of these modules. As a result, we also give an explicit description of submodule structures and obtain new simple non-weight modules over R. In particular, we recover some results about U ( h ) -free modules over the Lie algebra sl 2 obtained by J. Nilsson and over the Lie superalgebra osp ( 1 | 2 ) obtained by Y. Cai and K. Zhao.

Let A, B and C be artin algebras such that there is a recollement of D ( Mod A ) relative to D ( Mod B ) and D ( Mod C ) . We compare the algebras A, B and C with respect to syzygy-finite properties and Igusa-Todorov properties under suitable conditions and consider relevant results in the recollements of the bounded derived categories. Further, we characterize when the functor j ⁎ (resp., i ⁎ , i ! ) in a recollement ( D b ( mod B ) , D b ( mod A ) , D b ( mod C ) , i ⁎ , i ⁎ , i ! , j ! , j ⁎ , j ⁎ ) is an eventually homological isomorphism.

The $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) is a very useful combinatorial tool in graph isomorphism testing. We address the applicability of $k$-WL to recognition of graph properties. Let $G$ be an input graph with $n$ vertices. We show that, if $n$ is prime, then vertex-transitivity of $G$ can be seen in a straightforward way from the output of 2-WL on $G$ and on the vertex-individualized copies of $G$. However, if $n$ is divisible by 16, then $k$-WL is unable to distinguish between vertex-transitive and non-vertex-transitive graphs with $n$ vertices as long as $k=o(\sqrt n)$. Similar results are obtained for recognition of arc-transitivity., 30 pages, 2 figures. This paper supersedes Section 5 in the first version of arXiv:2002.04590. The 3rd version is a thorough revision of the paper

We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let S be a finite right (left) restriction Ehresmann semigroup whose corresponding Ehresmann category is an EI-category, that is, every endomorphism is an isomorphism. We show that the collection of finite right restriction Ehresmann semigroups whose categories are EI is a pseudovariety. We prove that the simple modules of the semigroup algebra k S (over any field k ) are formed by inducing the simple modules of the maximal subgroups of S via the corresponding Schutzenberger module. Moreover, we show that over fields with good characteristic the indecomposable projective modules can be described in a similar way but using generalized Green's relations instead of the standard ones. As a natural example, we consider the monoid PT n of all partial functions on an n-element set. Over the field of complex numbers, we give a natural description of its indecomposable projective modules and obtain a formula for their dimension. Moreover, we find certain zero entries in its Cartan matrix.

For any integers $m,n$ with $m\ne 0$ and $n>0$, let $G_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[z^m,x][z^n,y]\rangle$; for any integers $m,n>0$, let $H_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[x^m,z^n][y,z]\rangle$. By investigating cohomology jump loci of irreducible ${\rm GL}(2,\mathbb{C})$-character varieties, we show: if $m,m'\ne 0$, $n,n'>0$ and $G_{m',n'}\cong G_{m,n}$, then $m=m',n=n'$; if $m,m',n,n'>0$ and $H_{m',n'}\cong H_{m,n}$, then $m'=m, n'=n$., 18 pages

Let A be a commutative augmented ring and I be its augmentation ideal. This paper shows that the sequence { I n / I n + 1 } becomes stationary up to isomorphism. The result yields stability in the associated graded ring of A along I.

We prove that the isomorphism type of the subrack lattice of a finite group determines the nilpotence class. We analyze the problem of estimating the orders of the group elements corresponding to the atoms of the subrack lattice. As a result, we show that the subrack lattice determines p-nilpotence of the group if a certain condition is met.

A symbolic Lie p-ring presentation defines a family of nilpotent Lie rings with p n elements for infinitely many primes p and a fixed positive integer n. Symbolic Lie p-ring presentations are used in the classification of isomorphism types of nilpotent Lie rings of order p n for all primes p and n ≤ 7 . We describe an algorithm to compute the Schur multipliers of all nilpotent Lie rings in the family defined by a symbolic Lie p-ring presentation. We apply this to determine the Schur multipliers of all nilpotent Lie rings of order dividing p 6 for all primes p ≥ 5 . Via the Lazard correspondence this yields the Schur multipliers of all groups of order dividing p 6 for all primes p ≥ 5 .

A classical theorem of Veldkamp describes the center of an enveloping algebra of a Lie algebra of a semi-simple algebraic group in characteristic $p.$ We generalize this result to a class of Lie algebras with a property that they arise as the reduction modulo $p\gg 0$ from an algebraic Lie algebra $\mathfrak{g},$ such that $\mathfrak{g}$ has no nontrivial semi-invariants in $Sym(\mathfrak{g})$ and $Sym(\mathfrak{g})^{\mathfrak{g}}$ is a polynomial algebra. As an application, we solve the derived isomorphism problem of enveloping algebras for the above class of Lie algebras., Comment: Significantly revised, final version. To appear in Journal of Algebra

In this paper, we will introduce some types of Frechet fractional derivative defined on the class of linear correlated fuzzy-valued functions. Firstly, we study Frechet derivative and R-derivative of integer order and investigate their relationship with the well-known generalized Hukuhara derivatives in fuzzy metric space. Secondly, the Riemann-Liouville fractional integral of linear correlated fuzzy-valued functions is well-defined via an isomorphism between R 2 and subspace of fuzzy numbers space R F . That allows us to introduce three types of Frechet fractional derivatives, which are Frechet Caputo derivative, Frechet Riemann-Liouville derivative and Frechet Caputo-Fabrizio derivative. Moreover, some common properties of fuzzy Laplace transform for linear correlated fuzzy-valued function are investigated. Finally, some applications to fuzzy fractional differential equations are presented to demonstrate the usefulness of theoretical results.

In this work we approach three-dimensional evolution algebras from certain constructions performed on two-dimensional algebras. More precisely, we provide four different constructions producing three-dimensional evolution algebras from two-dimensional algebras. Also we introduce two parameters, the annihilator stabilizing index and the socle stabilizing index, which are useful tools in the classification theory of these algebras. Finally, we use moduli sets as a convenient way to describe isomorphism classes of algebras. Funding for open access charge: Universidad de Málaga / CBUA. Spanish Ministerio de Ciencia e Innovación through the project PID2019-104236GB-I00. Junta de Andalucía through the projects FQM-336 and UMA18-FEDERJA-119, all of them with FEDER funds. Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grant number 307910/2020-2 and Capes-PrInt grant number 88881.310538/2018-01 - Brazil.

Attribute reduction of rough set theory underlies knowledge acquisition and has two hierarchical types (classification-based and class-specific attribute reducts) and two perspectives from algebra and information theory; thus, there are four combined modes in total. Informational class-specific reducts are fundamental but lacking and are thus investigated by correspondingly constructing class-specific information measures. First, three types of information measures (i.e., information entropy, conditional entropy, and mutual information) are novelly established at the class level by hierarchical decomposition to acquire their hierarchical connection, systematical relationship, uncertainty semantics, and granulation monotonicity. Second, three types of informational class-specific reducts are correspondingly proposed to acquire their internal relationship, basic properties, and heuristic algorithm. Third, the informational class-specific reducts achieve their transverse connections, including the strength feature and consistency degeneration, with the algebraic class-specific reducts and their hierarchical connections, including the hierarchical strength and balance, with the informational classification-based reducts. Finally, relevant information measures and attribute reducts are effectively verified by decision tables and data experiments. Class-specific information measures deepen existing classification-based information measures by a hierarchical isomorphism, while the informational class-specific reducts systematically perfect attribute reduction by level and viewpoint isomorphisms; these results facilitate uncertainty measurement and information processing, especially at the class level.

While corporate venture capital funds (CVCs) are commonly analyzed as homogenous units, they display significant heterogeneity across various organizational aspects, which affect them and subsequently their portfolio firms. Using a sample of 383 European portfolio firms from the longitudinal VICO dataset, we first investigate the impact of investor type (independent vs corporate) on firm operating efficiency. We show that firms backed by CVCs suffer reductions in productivity. We then account for CVC heterogeneity and find that these significant reductions in operating efficiency only occur for ventures backed by endoisomorphistic CVCs, which resemble more corporate structures. By contrast, firms backed by exoisomorphistic CVCs, which resemble more independent venture capital structures, do not show significant differences in productivity compared to ventures that receive independent venture capital backing.

The aim of this paper is to compare the Cartier and covariant Dieudonne modules of connected p-divisible groups over perfect fields of positive characteristic p. In fact, we construct an explicit, canonical and functorial isomorphism and its inverse from the Dieudonne module to the Cartier module. The fact that the Cartier and (covariant) Dieudonne modules of a p-divisible formal Lie group over a perfect field are isomorphic is attributed to W. Messing, but a written proof does not exist in the literature ( [1, §5] ). We also give an application of this explicit isomorphism.

We prove a Thom isomorphism theorem for differential forms in the setting of transverse Lie algebra actions on foliated manifolds and foliated vector bundles., Comment: 29 pages. Contains material from first version of arXiv:1806.01908. Section 4 revised and expanded. Figure added

Structural Synthesis and Analysis plays a major role in identifying the Isomorphism and Symmetry of a Planetary Gear Train (PGT). In synthesis process, detection of isomorphism in PGTs with given number of links and Degree of Freedom (DOF) is essential. One gear pair and one turning pair will be added to the existing link to generate next level PGT. One isomorphic PGT is selected at random to generate higher link PGTs. Graph theory and matrix methods are used by different kinematicians to identify the symmetry and isomorphism in PGTs. Every method has its own merits and demerits. In this paper a comparison was made between the Hamming number technique and Modified Path Matrix (MPM) approach. For easy analysis, PGTs are converted in to graphs and matrices are extracted from these graphs. In this paper isomorphism is checked by using Adjacency matrix and Modified path Matrix. 4-link and 6-link PGTs were considered for analysis in both the methods and compared the results. It is observed that Hamming Number approach is simple and feasible method among the two to identify the symmetry and isomorphism in PGTs.

The Structural synthesis and topological analysis of Epicyclic gear trains (EGTs) have been great research area for checking isomorphic properties of mechanisms since few decades. In this paper, comparative analysis was done between three methods as Modified path matrix approach (MPM), Modified gradient method (MGM) and Bocher’s Technique (BT) for checking isomorphism in EGTs and its inversions. These methods have computed on the basis of reliability of results, computational application & structural properties for theirs merits and demerits by using MATLAB software. The methods have been compared with examples of 4-link EGTs and 6-link EGTs One-DOF. All examples have found satisfactory results on the basis of literature.

Almost equal molecular sizes of diiodoacetylene (C2I2) and the [Br3]− anion result in isomorphic neutral cocrystal (DMFc–C2I2) and [DMFc]+Br3− salt. This isomorphic pair of neutral and ionic crystals provides a unique example for the analysis of different types of bonding resulting in isotypic solid-state structures. Their energy frameworks visualize the obvious difference between their energetic structures: the lattice of DMFc–C2I2 is built upon the chains stabilized by I···(η5-C5Me5) halogen bonds; at the same time, [DMFc]+Br3− consists of electrostatically assembled layers associated by Br···π(Cp*) XBs interactions.

We will prove that there is a direct relationship between the Poincare subgroup of translations, and the group of tetrad transformations LB1 introduced in a previous paper. LB1 is the group composed of SO(1, 1) plus two kinds of discrete transformations. Translations have been extensively studied under the scope of gauge theories. By using the geometric structures built to prove this elementary result we will generalize it to the case of what we might call local translations. A special case of the latter is the Bondi–Metzner–Sachs subgroup of supertranslations. In order to accomplish this goal and since the group of translations is four-dimensional we will prove first that it is isomorphic to (⊗ LB1) 4 . In order to prove this claim we will introduce a system of differential equations involving several kinds of fields—Abelian, non-Abelian, spinor, gravitational. These fields will constitute the structure needed to build local tetrads of a new kind that allow for the proof to be carried out with simplicity. Results already obtained involving similar but not equal tetrads will be useful in our constructions and demonstrations. Translations and generalized translations isomorphic to tensor products of LB1 groups are not trivial results, because the LB1 group is composed of SO(1, 1) and two discrete transformations

We wish to fix an error in [1] . The list of representatives of the isomorphism classes of skew-symmetric products μ : g × g → g for a complex 3-dimensional vector space g given there in Proposition 3.5 corresponding to the rank-2 degenerate products characterized by (2) in Proposition 3.3, were collapsed into a single representative, whereas a one-parameter family of representatives should have been obtained, as it is well known from the classification of 3-dimensional Lie algebras. The mistake affects the statements of Propositions 3.5 and 5.4. This note gives the correct statements of these two results. The rest is unaffected.

If $\mathbb{F}$ is an ordered field and $M$ is a finite-rank torsion-free monoid, then one can embed $M$ into a finite-dimensional vector space over $\mathbb{F}$ via the inclusion $M \hookrightarrow \text{gp}(M) \hookrightarrow \mathbb{F} \otimes_{\mathbb{Z}} \text{gp}(M)$, where $\text{gp}(M)$ is the Grothendieck group of $M$. Let $\mathcal{C}$ be the class consisting of all monoids (up to isomorphism) that can be embedded into a finite-rank free commutative monoid. Here we investigate how the atomic structure and arithmetic properties of a monoid $M$ in $\mathcal{C}$ are connected to the combinatorics and geometry of its conic hull $\text{cone}(M) \subseteq \mathbb{F} \otimes_{\mathbb{Z}} \text{gp}(M)$. First, we show that the submonoids of $M$ determined by the faces of $\text{cone}(M)$ account for all divisor-closed submonoids of $M$. Then we appeal to the geometry of $\text{cone}(M)$ to characterize whether $M$ is a factorial, half-factorial, and other-half-factorial monoid. Finally, we investigate the cones of finitary, primary, finitely primary, and strongly primary monoids in $\mathcal{C}$. Along the way, we determine the cones that can be realized by monoids in $\mathcal{C}$ and by finitary monoids in $\mathcal{C}$., 40 pages, 4 figures; to appear in Linear Algebra and Its Applications

We continue the project started in [8] to describe the structure of the finite groups G of characteristic p in terms of their Baumann components and the conjugacy class Bau p ( G ) . The reduction theorem proved in [8] allows to assume that G has a unique Baumann component. In this paper we use this property to determine the isomorphism type of G / O p ( G ) and the action of G on Ω 1 ( Z ( O p ( G ) ) ) . In addition, we prove reduction theorems which allow to focus on groups G which satisfy G / O p ( G ) ≅ SL n ( q ) , Sp 2 n ( q ) or G 2 ( q ) and O p ( G ) ⩽ B for B ∈ Bau p ( G ) .

Given an integer n > 0 we give a computable injective listing of the isomorphism types of all computable abelian p-groups of Ulm type ≤n. We prove a similar result for certain classes of profinite groups.

In this paper, a set of generators (in a unique from) called the distinguished set of generators, of a cyclic code C of length n = 2 k (where k is a natural number) over Z 2 m is obtained. This set of generators is used to find the rank of the cyclic code C . It is proved that the rank of a cyclic code C of length n = 2 k over Z 2 m is equal to n − v , where v is the degree of a minimal degree polynomial in C . Then a description of all MHDR (maximum hamming distance with respect to rank) cyclic codes of length n = 2 k over Z 2 m is given. An example of the best cyclic codes over Z 8 of length 4 having largest minimum Hamming, Lee and Euclidean distances among all cyclic codes of the same rank is also given. Further, using an isomorphism between cyclic and negacyclic codes of odd length over finite chain rings given by Dinh, Lopez and Szabo, the above results are extended to cyclic and negacyclic codes of length p k over Z p m for an odd prime p .

On-line social networks, such as in Facebook and Twitter, are often studied from the perspective of friendship ties between agents in the network. Adversarial ties, however, also play an important role in the structure and function of social networks, but are often hidden. Underlying generative mechanisms of social networks are predicted by structural balance theory, which postulates that triads of agents, prefer to be transitive, where friends of friends are more likely friends, or anti-transitive, where adversaries of adversaries become friends. The previously proposed Iterated Local Transitivity (ILT) and Iterated Local Anti-Transitivity (ILAT) models incorporated transitivity and anti-transitivity, respectively, as evolutionary mechanisms. These models resulted in graphs with many observable properties of social networks, such as low diameter, high clustering, and densification. We propose a new, generative model, referred to as the Iterated Local Model (ILM) for social networks synthesizing both transitive and anti-transitive triads over time. In ILM, we are given a countably infinite binary sequence as input, and that sequence determines whether we apply a transitive or an anti-transitive step. The resulting model exhibits many properties of complex networks observed in the ILT and ILAT models. In particular, for any input binary sequence, we show that asymptotically the model generates finite graphs that densify, have clustering coefficient bounded away from 0, have diameter at most 3, and exhibit bad spectral expansion. We also give a thorough analysis of the chromatic number, domination number, Hamiltonicity, and isomorphism types of induced subgraphs of ILM graphs.

In this paper we study the combinatorial consequences of the relationship between rational Cherednik algebras of type G ( l , 1 , n ) , cyclic quiver varieties and Hilbert schemes. We classify and explicitly construct C ⁎ -fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles. Furthermore, we give a combinatorial description of the bijections between C ⁎ -fixed points induced by the Etingof-Ginzburg isomorphism and Nakajima reflection functors. We apply our results to obtain a new proof as well as a generalization of the q-hook formula.

We study and classify the 3-dimensional Hom-Lie algebras over $\mathbb{C}$. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space $\mathfrak{g}$. The well known Lie brackets for the 3-dimensional Lie algebras are included into appropriate isomorphism classes of such products representatives. For each product representative, we provide a complete set of canonical forms for the linear maps $\mathfrak{g} \to \mathfrak{g}$ that turn $g$ into a Hom-Lie algebra, thus characterizing the corresponding isomorphism classes. As by-products, Hom-Lie algebras for which the linear maps $\mathfrak{g} \to \mathfrak{g}$ are not homomorphisms for their products, are exhibited. Examples also arise of non-isomorphic families of HomLie algebras which share, however, a fixed Lie-algebra product on $\mathfrak{g}$. In particular, this is the case for the complex simple Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. Similarly, there are isomorphism classes for which their skew-symmetric bilinear products can never be Lie algebra brackets on $\mathfrak{g}$.

Let Δ be the set of all quadruples ( ν , μ , d , e ) of integers that satisfy 0 ≤ D − d 2 ≤ ν ≤ μ ≤ D − d ≤ D , e + d + D is even , | e | ≤ 2 ν − D + d , d ∈ { e + D − 2 ν , min ⁡ { D − μ , e + D − 2 ν + 2 ( N − 2 D ) } } . In [6] , it is shown for the Terwilliger algebra T of the Grassmann scheme J q ( N , D ) , N ≥ 2 D , that the isomorphism classes of irreducible T-modules W are determined by their endpoint ν, dual endpoint μ, diameter d, and auxiliary parameter e, that come from the Leonard system attached to W, and it is claimed without proof that the quadruples ( ν , μ , d , e ) belong to Δ, if d ≥ 1 . Let Λ be the set of triples ( α , β , ρ ) of non-negative integers that satisfy 0 ≤ α ≤ D − ρ 2 , 0 ≤ β ≤ N − D − ρ 2 , 0 ≤ α + β ≤ D − ρ . We construct a mapping from Λ to Δ which is bijective if N > 2 D and 2 : 1 if N = 2 D . We show that the set Λ naturally parameterizes the isomorphism classes of irreducible T-modules, by embedding the standard module of J q ( N , D ) in a bigger space that allows a U q ( s l ˆ 2 ) -module structure [9] . As a byproduct we have the following: for a fixed ρ, 0 ≤ ρ ≤ D , set N ′ = N − 2 ρ , D ′ = D − ρ , and Λ ρ = { ( α , β ) | ( α , β , ρ ) ∈ Λ } . Then Λ ρ is precisely the set that parameterizes the isomorphism classes of irreducible T-modules for the Johnson scheme J ( N ′ , D ′ ) [3] .

Two signed graphs are called switching isomorphic if one of them is isomorphic to a switching equivalent of the other. To determine the number of switching non-isomorphic signed graphs on a specific graph, we will establish a method based on the action of its automorphism group. As an application and computational result, we classify all the switching non-isomorphic signed graphs arising from the complete graph K 5 and the generalized Petersen graph GP ( 7 , 2 ) . The instances have minimum possible negative edges between switching equivalency classes. Moreover, some results on the automorphism groups of the target signed graphs are obtained.

In this paper, we construct a family of non-weight modules over the super-Virasoro algebras. Those modules when regarded as modules of the Ramond algebra and further restricted as modules over the Cartan subalgebra $\mathfrak{h}$ are free of rank $1$, while when regarded as modules of the Neveu-Schwarz algebra and further restricted as modules over the Cartan subalgebra $\mathfrak{H}$ are free of rank $2$. We obtain a sufficient and necessary condition for such modules to be simple. Moreover, we determine the isomorphism classes of these modules. Finally, we show that these modules constitute a complete classification of free $U(\mathfrak{h})$-modules of rank $1$ over the super-Virasoro algebra of Ramond type, and also constitute a complete classification of free $U(\mathfrak{H})$-modules of rank $2$ over the super-Virasoro algebra of Neveu-Schwarz type., Comment: 17 pages

We consider the algebra $\mathcal O(\mathsf M)$ of observables and the (formally) versal morphism $\eta: A \to \mathcal O(\mathsf M)$ defined by the noncommutative deformation functor $\mathsf{Def}_{\mathsf M}$ of a family $\mathsf M = \{ M_1, \dots, M_r \}$ of right modules over an associative $k$-algebra $A$. By the Generalized Burnside Theorem, due to Laudal, $\eta$ is an isomorphism when $A$ is finite dimensional, $\mathsf M$ is the family of simple $A$-modules, and $k$ is an algebraically closed field. The purpose of this paper is twofold: First, we prove a form of the Generalized Burnside Theorem that is more general, where there is no assumption on the field $k$. Secondly, we prove that the $\mathcal O$-construction is a closure operation when $A$ is any finitely generated $k$-algebra and $\mathsf M$ is any family of finite dimensional $A$-modules, in the sense that $\eta_B: B \to \mathcal O^B(\mathsf M)$ is an isomorphism when $B = \mathcal O(\mathsf M)$ and $\mathsf M$ is considered as a family of $B$-modules., Comment: 9 pages

We achieve an extremely useful description (up to isomorphism) of the Leavitt path algebra $L_K(E)$ of a finite graph $E$ with coefficients in a field $K$ as a direct sum of matrix rings over $K$, direct sum with a corner of the Leavitt path algebra $L_K(F)$ of a graph $F$ for which every regular vertex is the base of a loop. Moreover, in this case one may transform the graph $E$ into the graph $F$ via some step-by-step procedure, using the "source elimination" and "collapsing" processes. We use this to establish the main result of the article, that every nonzero corner of a Leavitt path algebra of a finite graph is isomorphic to a Leavitt path algebra. Indeed, we prove a more general result, to wit, that the endomorphism ring of any nonzero finitely generated projective $L_K(E)$-module is isomorphic to the Leavitt path algebra of a graph explicitly constructed from $E$. Consequently, this yields in particular that every unital $K$-algebra which is Morita equivalent to a Leavitt path algebra is indeed isomorphic to a Leavitt path algebra., 24 pages. Submitted for publication

There is a tight relation between the geometry of a directed graph and the algebraic structure of a Leavitt path algebra associated to it. In this note, we show a similar connection between the geometry of the graph and the structure of a certain monoid associated to it. This monoid is isomorphic to the positive cone of the graded K0-group of the Leavitt path algebra which is naturally equipped with a Z-action. As an example, we show that a graph has a cycle without an exit if and only if the monoid has a periodic element. Consequently a graph has Condition (L) if and only if the group Z acts freely on the monoid. We go on to show that the algebraic structure of Leavitt path algebras (such as simplicity, purely infinite simplicity, or the lattice of ideals) can be described completely via this monoid. Therefore an isomorphism between the monoids (or graded K0's) of two Leavitt path algebras implies that the algebras have similar algebraic structures. These all confirm that the graded Grothendieck group could be a sought-after complete invariant for the classification of Leavitt path algebras., 14 pages

A simple right alternative superalgebra is called singular if its even part has zero multiplication. The first example of a singular superalgebra was given in [1] . In [2] , a classification is given up to an isomorphism of 5-dimensional singular superalgebras. In this paper, the concepts of algebraically generated and linearly generated singular superalgebras are introduced. It is proved that an algebraically generated singular superalgebra with a 2-dimensional even part is linearly generated. It is also proved that every linearly generated singular superalgebra is isomorphic to a superalgebra associated with a suitable non-degenerate symmetric matrix. If the base field is algebraically closed, then for each admissible dimension of type 4 n + 1 ≥ 5 there is only one such superalgebra, called the standard one. Lie superalgebras of differentiations and automorphism groups of standard superalgebras are described. Examples of singular superalgebras in which the even and odd parts are infinite-dimensional are also indicated.

This work deals with the definability problem by quantifier-free first-order formulas over a finite algebraic structure. We show the problem to be coNP-complete and present a decision algorithm based on a semantical characterization of definable relations as those preserved by isomorphisms of substructures. Our approach also includes the design of an algorithm that computes the isomorphism type of a tuple in a finite algebraic structure. Proofs of soundness and completeness of the algorithms are presented, as well as empirical tests assessing their performances.

Motivated by the study of C-topology on lattice-ordered groups, Luan and Yang introduced the concept of filter topology on MV-algebras in a recent paper. In order to have a better perception of this concept, we study some basic facts and notable examples, including the standard MV-algebra and Chang's algebra, in this paper. Using the fact that an isomorphism between MV-algebras admits a topological isomorphism between filter topological MV-algebras, we show that every filter topology on a finite linearly ordered (finite simple) MV-algebra is discrete. In addition, we completely describe the shape of open sets in filter topology generated by a proper filter. Further, the description is applied to present the basic topological properties of a large class of filter topologies. As an application, we characterize an algebraic property of MV-algebras: an MV-algebra is bipartite if and only if it has a proper filter which generates a filter topology with exactly four open sets.

Multi-posets as the generalization of quantum B-algebras and partially ordered semigroups etc. were firstly introduced and studied by Rump. In this paper, we shall continue his work and investigate the completions of multi-posets and quantum B-algebras. Firstly, we introduce the notions of the (precoherent) completions of a multi-poset M and the (algebraic) consistent quantic nuclei on the power-set P ( M f ) of the free semigroup of M, and prove that the (precoherent) completions of a multi-poset M is completely determined by the (algebraic) consistent quantic nuclei on P ( M f ) up to isomorphism. Moreover, we give the concrete forms of the least and the largest standard (precoherent) completions and obtain the free quantale over a multi-poset in the category of multi-posets. Finally, when the completions and consistent quantic nuclei of multi-posets are applied to the cases of quantum B-algebras and partially ordered semigroups, several analogous results can also be obtained.

Counting problems whose solution is PORC were introduced in a famous paper by Higman (1960). We consider two specific counting problems with PORC solutions: the number of isomorphism types of d-generator class-2 Lie algebras over F q (as a function in q) and the number of isomorphism types of d-generator p-class 2 p-groups (as a function in p). We prove lower bounds for the degrees of their PORC formulae for all d ∈ N and we determine explicit PORC formulae for d ≤ 7 .

By an n-dimensional quadratic form over a field F (F-form) we mean a homogeneous quadratic polynomial in n variables with coefficients in F. We call two pairs of n-dimensional quadratic F-forms ( f 1 , g 1 ) , ( f 2 , g 2 ) isomorphic if there exists a nondegenerate linear change of variables taking ( f 1 , g 1 ) to ( f 2 , g 2 ) . We give a complete classification (up to isomorphism) of 3-dimensional pairs of forms ( f , g ) over an arbitrary field F of characteristic different from 2. We consider separately the cases of isotropic form f + t g (which is equivalent to existence of a common zero of f and g), and the anisotropic one. In the first case we make a classification depending on det ( f + t g ) , and in the second case depending on the even Clifford algebra C 0 ( f + t g ) and det ( f + t g ) .

We give an answer in the “geometric” setting to a question of [4] asking when local isomorphisms of k-schemes can be detected on the associated maps of local arc or jet schemes. In particular, we show that their ideal-closure operation a ↦ a ac (the arc-closure) on a local k-algebra ( R , m , L ) is trivial when R is Noetherian and k ↪ L is separable, and thus that such a germ Spec R has the (embedded) local isomorphism property.

Let L be a tensor product of two very ample line bundles on the smooth projective complex threefold X. Under the hypothesis that H 0 ( X , K X ( L ) ) ≠ 0 , we show that the restriction map r : Pic X → Pic Y is an isomorphism for very general Y in the linear system | L | . For such L this result recovers the Noether-Lefschetz theorem of Moishezon, who extended the original topological and Hodge-theoretic arguments of Lefschetz. We give a degeneration argument which is almost completely algebraic in nature.

Supervisory control synthesis is a method to automatically generate a correct-by-construction supervisory controller. Validation of the synthesized controller is an important step to guarantee correct and safe system behavior. Especially requirement validation for systems with numerous components can be a difficult and time-consuming task. This paper proposes a method that reduces the required validation time and effort of systems through symmetry reduction, and is based on the concept of isomorphism. Isomorphism of component models and requirement models means that these models are equivalent in behavior, and therefore only part of the system needs to be validated. This method is used in an industrial case study, in which a supervisory controller is synthesized for a road tunnel (the Koning Willem-Alexandertunnel, the Netherlands). In this case study, the modeling of the plant and the requirements, supervisor synthesis, simulation, and validation are described.

The identification of isomorphism among epicyclic gear trains (EGTs) and their mechanisms has great area of research for the last few years. In this paper, a novel approach based on Bocher’s equation is presented to identifying all distinct EGTs with different links and degree of freedoms. It is basically based on the connectivity matrices and the absolute sum of polynomial coefficient which identifying isomorphism in EGTs. Finally, the proposed method was examined on the basis of various examples from 4, 5, 6, and 8-links one-dof and 6-links two-dof EGTs. All examples have been found satisfactory results with existing literature.

International audience; For a system, a priori identifiability is a theoretical property depending only on the model and guarantees that its parameters can be uniquely determined from observations. This paper provides a survey of the various and numerous definitions of a priori identifiability given in the literature, for both deterministic continuous and discrete-time models. A classification is done by distinguishing analytical and algebraic definitions as well as local and global ones. Moreover, this paper provides an overview on the distinct methods to test the parameter identifiability. They are classified into the so-called output equality approaches, local state isomorphism approaches and differential algebra approaches. A few examples are detailed to illustrate the methods and complete this survey.

We give classifications of group gradings, up to equivalence and up to isomorphism, on the tensor product of a Cayley algebra $\mathcal{C}$ and a Hurwitz algebra over a field of characteristic different from 2. We also prove that the automorphism group schemes of $\mathcal{C}^{\otimes n}$ and $\mathcal{C}^n$ are isomorphic. On the other hand, we prove that the automorphism group schemes of a Smirnov algebra (a $35$-dimensional simple exceptional structurable algebra constructed from a Cayley algebra $\mathcal{C}$) and $\mathcal{C}$ are isomorphic. This is used to obtain classifications, up to equivalence and up to isomorphism, of the group gradings on Smirnov algebras., 29 pages

Detection of isomorphism in epicyclic gear trains (EGTs) and their mechanisms has been a great challenge for several researchers since few years. Checking for isomorphism at the conceptual stage will avoid duplication in the generation process. In this paper, an innovative approach based on Modified gradient method (MGM) is presented for identification of isomorphism of epicyclic gear trains and inversions of mechanisms. The new concept ‘MGM’ is proposed on basis of Train values and connectivity Matrices to be a quantitative measurement of isomorphism in EGTs. This is a novel, simple algorithm and less computation required to detect the duplication of EGTs. The proposed algorithm is applied to various examples from 4-links, 5-links, 6-links, and 8-links 1-DOF of the epicyclic gear trains. All the examples have been found satisfactory results with existing literature.

We present a complete Coxeter spectral classification of finite one-peak posets I = T ∪ { ⁎ } that are principal in the sense that their symmetric Gram matrix G I : = 1 2 ( C I + C I t r ) ∈ M | I | ( Q ) is positive semi-definite of rank | I | − 1 , where C I ∈ M | I | ( Z ) is the incidence matrix of I; equivalently: posets T with the quadratic Tits form q T : Z | T | + 1 → Z (in the sense of Drozd) non-negative and the kernel Ker q T : = { v ∈ Z | T | + 1 ; q T ( v ) = 0 } ⊆ Z | T | + 1 infinite cyclic. We continue the study started in M. Gąsiorek, D. Simson (2012) [11] . We show that, up to the strong Gram Z -congruence and the weak Gram Z -congruence, every one-peak principal poset is uniquely determined by its complex Coxeter spectrum specc I ⊆ C . Moreover, we calculate the exact number of all such posets of a given size (up to isomorphism) and present their Hasse quivers. We finish the paper with conjecture on the exact number of principal (positive) posets. According to our knowledge no formula is known for the number of non-isomorphic posets in a general case.