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The authors developed in a recent paper natural dualities for finitely generated quasivarieties of Sugihara algebras. They thereby identified the admissibility algebras for these quasivarieties which, via the Test Spaces Method devised by Cabrer et al., give access to a viable method for studying admissible rules within relevance logic, specifically for extensions of the deductive system R-mingle. This paper builds on the work already done on the theory of natural dualities for Sugihara algebras. Its purpose is to provide an integrated suite of multisorted duality theorems of a uniform type, encompassing finitely generated quasivarieties and varieties of both Sugihara algebras and Sugihara monoids, and embracing both the odd and the even cases. The overarching theoretical framework of multisorted duality theory developed here leads on to amenable representations of free algebras. More widely, it provides a springboard to further applications. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of small and big Ramsey degrees of classes of finite unary algebras in an arbitrary (not necessarily finite) algebraic language Ω. Our results generalize some Ramsey-type results of M. Sokić concerning finite unary algebras over finite languages. To do so, we develop a completely new strategy that relies on the fact that right adjoints preserve the Ramsey property. We then treat unary algebras as Eilenberg-Moore coalgebras for a functor with comultiplication, and using pre-adjunctions transport the Ramsey properties, we are interested in from the category of finite or countably infinite chains of order type ω. Moreover, we show that finite objects have finite big Ramsey degrees in the corresponding cofree structures over countably many generators. [ABSTRACT FROM AUTHOR]

Let 퓟n := H*((ℝP∞)n) ≅ ℤ2[x1, x2, ..., xn] be the graded polynomial algebra over ℤ2, where ℤ2 denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra 퓟n, viewed as a graded left module over the mod-2 Steenrod algebra, 퓐. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. In this article, we study the hit problem for the case n = 6 in the generic degree dr = 6(2r − 1) + 4.2r with r an arbitrary non-negative integer. By considering ℤ2 as a trivial 퓐-module, then the hit problem is equivalent to the problem of finding a basis of ℤ2-vector space ℤ2 ⊗퓐퓟n. The main goal of the current article is to explicitly determine an admissible monomial basis of the ℤ2 vector space ℤ2 ⊗퓐퓟6 in some degrees. As an application, the behavior of the sixth Singer algebraic transfer in the degree 6(2r − 1) + 4.2r is also discussed at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we define and study the variety of tense modal pseudocomplemented De Morgan algebras. This variety is a proper subvariety of the variety of tense tetravalent modal algebras. A tense modal pseudocomplemented De Morgan algebra is a modal pseudocomplemented De Morgan algebra endowed with two tense operators G and H satisfying additional conditions. Also, the variety of tense modal pseudocomplemented De Morgan algebras is intimately connected with some well-known varieties of De Morgan algebras with tense operators. [ABSTRACT FROM AUTHOR]

In 2012, the second author introduced, and initiated the investigations into, the variety 퓘 of implication zroupoids that generalize De Morgan algebras and ∨-semilattices with 0. An algebra A = 〈 A, →, 0 〉, where → is binary and 0 is a constant, is called an implication zroupoid (퓘-zroupoid, for short) if A satisfies: (x → y) → z ≈ [(z′ → x) → (y → z)′]′, where x′ := x → 0, and 0″ ≈ 0. Let 퓘 denote the variety of implication zroupoids and A ∈ 퓘. For x, y ∈ A, let x ∧ y := (x → y′)′ and x ∨ y := (x′ ∧ y′)′. In an earlier paper, we had proved that if A ∈ 퓘, then the algebra Amj = 〈A, ∨, ∧〉 is a bisemigroup. The purpose of this paper is two-fold: First, we generalize the notion of semidistributivity from lattices to bisemigroups and prove that, for every A ∈ 퓘, the bisemigroup Amj is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety 퓜퓔퓙 of 퓘, defined by the identity: x ∧ y ≈ x ∨ y, satisfies the Whitman Property. We conclude the paper with two open problems. [ABSTRACT FROM AUTHOR]

In 1973, Katriňák proved that regular double p-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heyting implication and its dual in terms of pseudocomplement and its dual. In this paper, we prove a converse to Katriňák's theorem, in the sense that in the variety ℝ D ℙ ℂ ℍ of regular dually pseudocomplemented Heyting algebras, the implication operation → satisfies Katriňák's formula. As applications of this result together with the above-mentioned Katriňák's theorem, we show that the varieties ℝ D B L ℙ , ℝ D ℙ ℂ ℍ , ℝ ℙ ℂ ℍ d and ℝ D B L ℍ of regular double p-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties ℝ D M ℙ , ℝ D M ℍ , ℝ D M D B L ℍ , ℝ D M D B L ℙ of regular De Morgan p-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double p-algebras, respectively, are also term-equivalent to each other. From these results and recent results of Adams, Sankappanavar and Vaz de Carvalho on varieties of regular double p-algebras and regular pseudocomplemented De Morgan algebras, we deduce that the lattices of subvarieties of all these varieties have cardinality 2 ℵ 0 . We then define new logics, ℛ D P C ℋ , ℛ P C ℋ d and ℛ D ℳ ℋ , and show that they are algebraizable with ℝ D ℙ ℂ ℍ , ℝ ℙ ℂ ℍ d and ℝ D M ℍ , respectively, as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality 2 ℵ 0 . [ABSTRACT FROM AUTHOR]

Let W ⁢ (n) be the Jacobson–Witt algebra over algebraic closed field 핂 with characteristic p > 2 . In [K. Ou and B. Shu, Borel subalgebras of restricted Cartan-type Lie algebras, J. Algebra Appl. 21 2022, 11, Paper No. 2250210], we introduced the so-called B-subalgebra of W ⁢ (n) , which serves as an analog of the Borel subalgebra of classical Lie algebras. As a sequel, we describe the structure of the variety ℬ consisting of all B-subalgebras of W ⁢ (n) in this paper. This variety presents an analog of the flag variety for classical Lie algebras. It is shown that ℬ is related to the variety of all full flags in 핂 n + 1 . Additionally, we provide a detailed description of the varieties for W ⁢ (1) as an illustrative example. With the above setting-up, one may establish the Springer theory and geometric representations for the Jacobson–Witt algebras. [ABSTRACT FROM AUTHOR]

In this paper, we continue the discussion about relations between exponential polynomials and generalized moment functions on a commutative hypergroup. We are interested in the following problem: is it true that every finite-dimensional variety is spanned by moment functions? Let m be an exponential on X. In our former paper, we have proved that if the linear space of all m-sine functions in the variety of an m-exponential monomial is (at most) one-dimensional, then this variety is spanned by moment functions generated by m. In this paper, we show that this may happen also in cases where the m-sine functions span a more than one-dimensional subspace in the variety. We recall the notion of a polynomial hypergroup in d variables, describe exponentials on it and give the characterization of the so-called m-sine functions. Next we show that the Fourier algebra of a polynomial hypergroup in d variables is the polynomial ring in d variables. Finally, using the Ehrenpreis–Palamodov Theorem, we show that every exponential polynomial on the polynomial hypergroup in d variables is a linear combination of moment functions contained in its variety. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new technique in the study of the * -regular closure of some specific group algebras KG inside 풰 ⁢ (G) , the * -algebra of unbounded operators affiliated to the group von Neumann algebra 풩 ⁢ (G) . The main tool we use for this study is a general approximation result for a class of crossed product algebras of the form C K ⁢ (X) ⋊ T ℤ , where X is a totally disconnected compact metrizable space, T is a homeomorphism of X, and C K ⁢ (X) stands for the algebra of locally constant functions on X with values on an arbitrary field K. The connection between this class of algebras and a suitable class of group algebras is provided by the Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family of ℓ 2 -Betti numbers arising from the lamplighter group, most of them transcendental. [ABSTRACT FROM AUTHOR]

In this paper, we study representations of non-finitely graded Lie algebras 풲 ⁢ ( ϵ ) {\mathcal{W}(\epsilon)} related to Virasoro algebra, where ϵ = ± 1 {\epsilon=\pm 1} . Precisely speaking, we completely classify the free 풰 ⁢ ( 픥 ) {\mathcal{U}(\mathfrak{h})} -modules of rank one over 풲 ⁢ ( ϵ ) {\mathcal{W}(\epsilon)} , and find that these module structures are rather different from those of other graded Lie algebras. We also determine the simplicity and isomorphism classes of these modules. [ABSTRACT FROM AUTHOR]

We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies Λ ⁢ (X m , n) ⊂ E of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial. In particular, we obtain a formula for ∇ m s λ which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one. [ABSTRACT FROM AUTHOR]

The aim of the paper is to provide a characterization criterion of exponential solvable Frobenius Lie algebras (having open coadjoint orbits), in terms of primitive ideals of the associated enveloping algebra. In the case of complex solvable Lie algebras, we also show that an algebraic adjoint orbit is open if and only if the associated primitive ideal through the Dixmier map is trivial. [ABSTRACT FROM AUTHOR]

Hecke–Kiselman algebras A Θ , over a field 핂 , associated to finite oriented graphs Θ are considered. It has been known that every such algebra is an automaton algebra in the sense of Ufranovskii. In particular, its Gelfand–Kirillov dimension is an integer if it is finite. In this paper, a numerical invariant of the graph Θ that determines the dimension of A Θ is found. Namely, we prove that the Gelfand–Kirillov dimension of A Θ is the sum of the number of cyclic subgraphs of Θ and the number of oriented paths of a special type in the graph, each counted certain specific number of times. [ABSTRACT FROM AUTHOR]

The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety 퓔6(핂) over an arbitrary field 핂. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions 핆' over 핂 (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties are linked to the Freudenthal–Tits magic square, and discuss how they would even fit in, when also allowing the sextonions and other "degenerate composition algebras" as the algebras used to construct the square. [ABSTRACT FROM AUTHOR]

In this paper, a new categorical crystal structure for the quantum affine algebras is presented. We introduce the notion of extended crystals B ^ 픤 ⁢ (∞) for an arbitrary quantum group U q ⁢ (픤) , which is the product of infinite copies of the crystal B ⁢ (∞) . For a complete duality datum 풟 in the Hernandez–Leclerc category 풞 픤 0 of a quantum affine algebra U q ′ ⁢ (픤) , we prove that the set ℬ 풟 ⁢ (픤) of the isomorphism classes of simple modules in 풞 픤 0 has an extended crystal structure isomorphic to B ^ 픤 fin ⁢ (∞) . An explicit combinatorial description of the extended crystal ℬ 풟 ⁢ (픤) for affine type A n (1) is given in terms of affine highest weights. [ABSTRACT FROM AUTHOR]

The paper contributes to Majorana theory. Among the eight non-trivial Norton–Sakuma algebras, four algebras are closed on the set of Majorana generators. These algebras are 2A, 2B, 3C and 4B. The classification of Majorana representations restricted to the closed shapes was anticipated for a long time. In the present article the classification is achieved for shapes restricted to 2A, 3C and 4B and for the set of generating involutions in the target group forming a single conjugacy class. Timmesfeld's classification of {3, 4}+-transposition groups reduces to consideration of just three groups: L3(2), G2(2)' and 3D4(2). Each of these groups possesses a unique Majorana representation of the required shape. Only the representation of L3(2), known before, is based on an embedding into the Monster. [ABSTRACT FROM AUTHOR]

The paper deals with the monounary algebras for which the second centralizer equals the first centralizer. We describe Green's relations on the semigroup C, where C is the centralizer of such algebra. [ABSTRACT FROM AUTHOR]

We investigate the role of gentle algebras in higher homological algebra. In the first part of the paper, we show that if the module category of a gentle algebra Λ contains a d-cluster tilting subcategory for some d ≥ 2 , then Λ is a radical square zero Nakayama algebra. This gives a complete classification of weakly d-representation finite gentle algebras. In the second part, we use a geometric model of the derived category to prove a similar result in the triangulated setup. More precisely, we show that if 풟 b ⁡ (Λ) contains a d-cluster tilting subcategory that is closed under [ d ] , then Λ is derived equivalent to an algebra of Dynkin type A. Furthermore, our approach gives a geometric characterization of all d-cluster tilting subcategories of 풟 b ⁡ (Λ) that are closed under [ d ] . [ABSTRACT FROM AUTHOR]

The broadly applied notions of Lie bialgebras, Manin triples, classical r-matrices and 풪 -operators of Lie algebras owe their importance to the close relationships among them. Yet these notions and their correspondences are mostly understood as classes of objects and maps among the classes. To gain categorical insight, this paper introduces, for each of the classes, a notion of homomorphisms, uniformly called coherent homomorphisms, so that the classes of objects become categories and the maps among the classes become functors or category equivalences. For this purpose, we start with the notion of an endo Lie algebra, consisting of a Lie algebra equipped with a Lie algebra endomorphism. We then generalize the above classical notions for Lie algebras to endo Lie algebras. As a result, we obtain the notion of coherent endomorphisms for each of the classes, which then generalizes to the notion of coherent homomorphisms by a polarization process. The coherent homomorphisms are compatible with the correspondences among the various constructions, as well as with the category of pre-Lie algebras. [ABSTRACT FROM AUTHOR]

We prove duality results for residual intersections that unify and complete results of van Straten, Huneke–Ulrich and Ulrich, and settle conjectures of van Straten and Warmt. Suppose that I is an ideal of codimension g in a Gorenstein ring, and J ⊂ I is an ideal with s = g + t generators such that K := J : I has codimension s. Let I ¯ be the image of I in R ¯ := R / K. In the first part of the paper we prove, among other things, that under suitable hypotheses on I, the truncated Rees ring R ¯ ⊕ I ¯ ⊕ ⋯ ⊕ I ¯ t + 1 is a Gorenstein ring, and that the modules I ¯ u and I ¯ t + 1 - u are dual to one another via the multiplication pairing into I ¯ ≅ t + 1 ωR ¯. In the second part of the paper we study the analogue of residue theory, and prove that, when R / K is a finite-dimensional algebra over a field of characteristic 0 and certain other hypotheses are satisfied, the socle of I t + 1 / J ⁢ I t ≅ ωR/K is generated by a Jacobian determinant. [ABSTRACT FROM AUTHOR]

Let F be an algebraically closed field and let R be a locally finite algebra over F. This paper aims to show that any element of R is a product of at most three unipotent elements from R if and only if the element lies in the first derived subgroup of the unit group of R. In addition, this necessary and sufficient condition is applied to twisted group algebras of locally finite groups over a field of either zero characteristic or characteristic p ≠ 0 for some prime p. Moreover, we explore some crucial properties satisfied by certain algebras like the conditions concerned with the connections between unipotent elements of index 2 and commutators, as well as we investigate the unipotent radical of some subgroups of a finite-dimensional algebra R over a field F with at least four elements. In particular, we again apply these results to twisted group algebras. [ABSTRACT FROM AUTHOR]

The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities. [ABSTRACT FROM AUTHOR]

Let R be an associative ring. A multiplicative semi-derivation d is a map on R satisfying d ⁢ (x ⁢ y) = d ⁢ (x) ⁢ g ⁢ (y) + x ⁢ d ⁢ (y) = d ⁢ (x) ⁢ y + g ⁢ (x) ⁢ d ⁢ (y) and d ⁢ (g ⁢ (x)) = g ⁢ (d ⁢ (x)) {d(xy)=d(x)g(y)+xd(y)=d(x)y+g(x)d(y)\quad\text{and}\quad d(g(x))=g(d(x))} for all x , y ∈ R {x,y\in R} , where g is any map on R. In this paper, we have obtained some conditions on R, which make d additive. Finally, we have also shown that every multiplicative semi-derivation on M n ⁢ (ℂ) {M_{n}(\mathbb{C})} , the algebra of all n × n {n\times n} matrices over the field ℂ {\mathbb{C}} of complex numbers, is an additive derivation. [ABSTRACT FROM AUTHOR]

In this paper, we study varieties and characterize convolution operators on the algebras related to the new structures of KPC-hypergroups, which are a generalization of the classical ones. [ABSTRACT FROM AUTHOR]

Let f : ℝ n → ℝ {f\colon\mathbb{R}^{n}\rightarrow\mathbb{R}} be a polynomial and 𝒵 ⁢ (f) {\mathcal{Z}(f)} its zero set. In this paper, in terms of the so-called Newton polyhedron of f, we present a necessary criterion and a sufficient condition for the compactness of 𝒵 ⁢ (f) {\mathcal{Z}(f)}. From this we derive necessary and sufficient criteria for the stable compactness of 𝒵 ⁢ (f) {\mathcal{Z}(f)}. [ABSTRACT FROM AUTHOR]

This paper is the complementary work of [S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory 10 2016, 3, 451–532]. Ramified quadratic extensions E / F {E/F} , where F is a finite unramified field extension of ℚ 2 {\mathbb{Q}_{2}} , fall into two cases that we call Case 1 and Case 2. In our previous work, we obtained the local density formula for a ramified hermitian lattice in Case 1. In this paper, we obtain the local density formula for the remaining Case 2, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with [W. T. Gan and J.-K. Yu, Group schemes and local densities, Duke Math. J. 105 2000, 3, 497–524] and our previous work, allows the computation of the mass formula for any hermitian lattice (L , H) {(L,H)} , when a base field is unramified over ℚ {\mathbb{Q}} at a prime (2) {(2)}. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group (not necessarily abelian) and B be a homogeneous Banach space on G, which is in a good situation with respect to a homogeneous function algebra on G. Feichtinger showed that there exists a minimal Banach space Bmin in the family of all homogenous Banach spaces C on G, containing all elements of B with compact support. In this paper, the amenability and super amenability of Bmin with respect to the convolution product or with respect to the pointwise product are showed to correspond to amenability, discreteness or finiteness of the group G and conversely. We also prove among other things that Bmin is a symmetric Segal subalgebra of L1(G) on an IN-group G, under certain conditions, and we apply our results to study pseudo-amenability and some other homological properties of Bmin on IN-groups. Furthermore, we determine necessary and sufficient conditions on A under which Amin with the pointwise product is an abstract Segal algebra or Segal algebra in A, whenever A is a homogeneous function algebra with an approximate identity. We apply these results to study amenability of some Feichtinger algebras with respect to the pointwise product. [ABSTRACT FROM AUTHOR]

Let ℜ be a ring with center Z(ℜ). In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation 𝔉 such that ([𝔉(sm)sn + sn𝔉(sm), sr]k)l ∈ Z(ℜ) or every s ∈ ℜ, then either 𝔉(s) = ps for every s ∈ ℜ or ℜ satisfies s4 and 𝔉(s) = sp for every s ∈ ℜ and p ∈ 𝔘, the Utumi quotient ring of ℜ. As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide a cohomology of n-Hom-Leibniz algebras. Moreover, we study some higher operations on cohomology spaces and deformations. [ABSTRACT FROM AUTHOR]

Let G be a group, let A = ⊕ g ∈ G A g be an epsilon-strongly graded ring over G, let R := A 1 be the homogeneous component associated with the identity of G, and let 홿횒회횂 ⁢ (R) be the Picard semigroup of R. In the first part of this paper, we prove that the isomorphism class [ A g ] is an element of 홿횒회횂 ⁢ (R) for all g ∈ G . Moreover, the association g ↦ [ A g ] determines a partial representation of G on 홿횒회횂 ⁢ (R) which induces a partial action γ of G on the center Z ⁢ (R) of R. Sufficient conditions for A to be an Azumaya R γ -algebra are presented if R is commutative. In the second part, we study when B is a partial crossed product in the following cases: B = M n ⁡ (A) is the ring of matrices with entries in A, or B = END A (M) = ⊕ l ∈ G Mor A (M , M) l is the direct sum of graded endomorphisms of graded left A-modules M with degree l, or B = END A ⁡ (M) where M = A ⊗ R N is the induced module of a left R-module N. Assuming that R is semiperfect, we prove that there exists a subring of A which is an epsilon-strongly graded ring over a subgroup of G and it is graded equivalent to a partial crossed product. [ABSTRACT FROM AUTHOR]

In this paper, a class of non-weight modules over N = 1 Lie superalgebras of Block type (without center) is studied. These modules over Ramond algebras of Block type are free of rank 1 when restricted to the universal enveloping algebra of the Cartan subalgebra, while such modules are free of rank 2 over the Neveu–Schwarz algebras of Block type. We present the classification of these modules. Furthermore, the simplicity of these modules is determined. [ABSTRACT FROM AUTHOR]

This article is devoted to the local geometry of everywhere 2-nondegenerate CR manifolds M of hypersurface type. An absolute parallelism for such structures was recently constructed independently by Isaev and Zaitsev, Medori and Spiro, and Pocchiola in the minimal possible dimension (dimM = 5), and for dimM = 7 in certain cases by the first author. In the present paper, we develop a bigraded (i.e., ℤ × ℤ-graded) analog of Tanaka's prolongation procedure to construct an absolute parallelism for these CR structures in arbitrary (odd) dimension with Levi kernel of arbitrary admissible dimension. We introduce the notion of a bigraded Tanaka symbol – a complex bigraded vector space – containing all essential information about the CR structure. Under the additional regularity assumption that the symbol is a Lie algebra, we define a bigraded analog of the Tanaka universal algebraic prolongation, endowed with an anti-linear involution, and prove that for any CR structure with a given regular symbol there exists a canonical absolute parallelism on a bundle whose dimension is that of the bigraded universal algebraic prolongation. Moreover, we show that for each regular symbol there is a unique (up to local equivalence) such CR structure whose algebra of infinitesimal symmetries has maximal possible dimension, and the latter algebra is isomorphic to the real part of the bigraded universal algebraic prolongation of the symbol. In the case of 1-dimensional Levi kernel we classify all regular symbols and calculate their bigraded universal algebraic prolongations. In this case, the regular symbols can be subdivided into nilpotent, strongly non-nilpotent, and weakly non-nilpotent. The bigraded universal algebraic prolongation of strongly non-nilpotent regular symbols is isomorphic to the complex orthogonal algebra 𝔰 ⁢𝔬(m, ℂ), where m = 1/2 ⁢ (dimM + 5). Any real form of this algebra – except 𝔰𝔬(m) and 𝔰𝔬(m−1, 1) – corresponds to the real part of the bigraded universal algebraic prolongation of exactly one strongly non-nilpotent regular CR symbol. However, for a fixed dimM ≥ 7 the dimension of the bigraded universal algebraic prolongations of all possible regular CR symbols achieves its maximum on one of the nilpotent regular symbols, and this maximal dimension is 1/4(dim m−1)2 + 7. [ABSTRACT FROM AUTHOR]

In this paper we define and study triangulated categories in which the Homspaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi--Yau type property. If 풞 is such a category, we say that 풞 is Calabi--Yau with dim 풞 ≤ 1. We extend the notion of Calabi--Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categories CM R of Cohen--Macaulay modules. We explain the connection between Calabi--Yau reduction of CM R and both partial crepant resolutions and ℚ-factorial terminalizations of Spec R, and we show under quite general assumptions that Calabi--Yau reductions exist. In the remainder of the paper we focus on complete local c An singularities R. By using a purely algebraic argument based on Calabi--Yau reduction of CM R, we give a complete classification of maximal modifying modules in terms of the symmetric group, generalizing and strengthening results in [7, 10], where we do not need any restriction on the ground field. We also describe the mutation of modifying modules at an arbitrary (not necessarily indecomposable) direct summand. As a corollary when k = ℂ we obtain many autoequivalences of the derived category of the ℚ-factorial terminalizations of Spec R. [ABSTRACT FROM AUTHOR]

Let A be a unital algebra, δ be a linear mapping from A into itself and m, n be fixed integers. We call δ an (m, n)-derivable mapping at Z, if mδ(AB) + nδ(BA) = mδ(A)B + mAδ(B) + nδ(B)A for all A,B ∈ A with AB = Z. In this paper, (m, n)-derivable mappings at 0 (resp. IA ⊕ 0, I) on generalized matrix algebras are characterized. We also study (m, n)-derivable mappings at 0 on CSL algebras. We reveal the relationship between this kind of mappings with Lie derivations, Jordan derivations and derivations. [ABSTRACT FROM AUTHOR]

This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ (U , V) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , (λ , μ) ↦ 𝔏 ⁢ (λ , μ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ (U , V) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]). [ABSTRACT FROM AUTHOR]

The present paper deals with the Lie triple systems via Leibniz algebras. A perfect Lie algebra as a perfect Leibniz algebra and as a perfect Lie triple system is considered and the appropriate universal central extensions are studied. Using properties of Leibniz algebras, it is shown that the Lie triple system universal central extension is either the universal central extension of the Leibniz algebra or the universal central extension of the Lie algebra. [ABSTRACT FROM AUTHOR]

In this paper, we study (non-Abelian) extensions of a given hom-Lie color algebra and provide a geometrical interpretation of extensions. In particular, we characterize an extension of a hom-Lie color algebra 𝔤 by another hom-Lie color algebra 𝔥 and discuss the case where 𝔥 has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, curvature and the Bianchi identity for possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie color algebras, i.e., we show that in order to have an extendible hom-Lie color algebra, there should exist a trivial member of the third cohomology. [ABSTRACT FROM AUTHOR]

In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer [Quotients of twisted Bruhat orders, J. Algebra 163 1994, 3, 861–879]. We also show that, for a non-finite and non-cofinite biclosed set B in the positive system of an affine root system with rank greater than 2, the set of elements having a fixed B-twisted length is infinite. This implies that the twisted strong and weak Bruhat orders have an infinite antichain in those cases. Finally, we show that twisted weak Bruhat order can be applied to the study of the tope poset of an infinite oriented matroid arising from an affine root system. [ABSTRACT FROM AUTHOR]

Recently, the problem of bounding the sup norms of L2-normalized cuspidal automorphic newforms ϕ on GL2 in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to ∥ϕ∥∞ ≪ N1/4 + ϵ, at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound ∥ϕ∥∞ ≪ N1/2 + ϵ in this setup (due to [A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 2017, 1009–1045]) and matches a lower bound due to [N. Templier, Large values of modular forms, Camb. J. Math. 2 2014, 1, 91–116], thus our result is essentially optimal in this case. [ABSTRACT FROM AUTHOR]

In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the 2 × 2 lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ. [ABSTRACT FROM AUTHOR]

In this paper we define the monadic pseudo BE-algebras and investigate their properties. We prove that the existential and universal quantifiers of a monadic pseudo BE-algebra form a residuated pair. Special properties are studied for the particular case of monadic bounded commutative pseudo BE-algebras. Monadic classes of pseudo BE-algebras are investigated and it is proved that the quantifiers on bounded commutative pseudo BE-algebras are also quantifiers on the corresponding pseudo MV-algebras. The monadic deductive systems and monadic congruences of monadic pseudo BE-algebras are defined and their properties are studied. It is proved that, in the case of a monadic distributive commutative pseudo BE-algebra there is a one-to-one correspondence between monadic congruences and monadic deductive systems, and the monadic quotient pseudo BE-algebra algebra is also defined. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the class of decomposable double MS-algebras. Necessary and sufficient conditions for a decomposable MS-algebra to be a decomposable double MS-algebra are deduced. We construct decomposable double MS-algebras by means of decomposable MS-quadruples and we prove that there exists a one-to-one correspondence between decomposable double MS-algebras and decomposable MS-quadruples. Moreover, a construction of decomposable K2-algebras (Stone algebras) by means of K2-quadruples (Stone quadruples) is given. We conclude by introducing and characterizing isomorphisms of decomposable double MS-algebras in terms of decomposable MS-quadruples. [ABSTRACT FROM AUTHOR]

R. W. Carey and J. Pincus in [] proposed an index theory for non-Fredholm bounded operators T on a separable Hilbert space such that is in the trace class. We showed in [] using Dirac-type operators acting on sections of bundles over that we could construct bounded operators T satisfying the more general condition that the operator is in the trace class. We proposed there a 'homological index' for these Dirac-type operators given by . In this paper we show that the index introduced in [] represents the result of a paring between a cyclic homology theory for the algebra generated by T and and its dual cohomology theory. This leads us to establish the homotopy invariance of our homological index (in the sense of cyclic theory). We are then able to define in a very general fashion a homological index for certain unbounded operators and prove invariance of this index under a class of unbounded perturbations. [ABSTRACT FROM AUTHOR]

In this paper we shall find some sufficient conditions for a uniqueness polynomial to be a strong uniqueness polynomial, as this type of problemwas never investigated by researchers earlier.We also exhibit some examples to substantiate our theorems. [ABSTRACT FROM AUTHOR]

This paper presents algebraic transformations of cadastral topological data. The traditional notation of data in attribute tables has been transformed into a matrix. The new approach supports algebraic transformations and has produced new descriptive forms for cadastral structures. In the second part of the paper, new topological data were transformed to a quantum metric to generate new descriptions of the neighborhood of sets of geometric elements relating to cadastral data. The resulting descriptions can be used in analytical processes such as neighborhood analyses and aggregation of areas. [ABSTRACT FROM AUTHOR]

In this paper we generalize and unify several results proved in recent papers about the existence of normal p-complements and conditions on supersolubility. Moreover a counterexample is given to a question in [Guo and Wei, J. Group Theory 13: 267-276, 2010] and it is proved that a finite group is 2-nilpotent if the cyclic subgroups of order less or equal than four are strongly closed. [ABSTRACT FROM AUTHOR]

In an earlier paper we proved some results concerning Gross's conjecture on tori. This conjecture, which we call the Minus Conjecture, is closely related to a conjecture of Burns, which is now known to hold generally in the absolutely abelian setting; however Burns' conjecture does not directly imply the Minus Conjecture. The result proved in the earlier paper was concerned with imaginary absolutely abelian extensions K/ℚ of the form K = FK+, with F imaginary quadratic and K+/ℚ being tame, l-elementary and ramified at most at two primes. In the present paper we complement these results by proving the Minus Conjecture for extensions K/ℚ as above but without any restriction on the number s of ramified primes. The price we have to pay for this generality is that our proof only works if the odd prime l is large enough, more precisely if l ≧ 3( s + 1). There is one more restriction, namely . (A similar restriction was already needed in our previous paper.) [ABSTRACT FROM AUTHOR]

Throughout this paper we show that under certain conditions the method for describing solvable Leibniz (resp. Lie) algebras with given nilradical by means of non-nilpotent outer derivations of the nilradical is also applicable to the case Leibniz (resp. Lie) superalgebras. Moreover, after having established the general method for Lie and Leibniz superalgebras, we classify all the solvable superalgebras on a very important class of each of them, that is, those with nilradical of maximal nilindex. Note that for (n + m) {(n+m)} -dimensional superalgebras this maximal nilindex is n + m - 1 {n+m-1} in the Lie case and n + m {n+m} in Leibniz. [ABSTRACT FROM AUTHOR]