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In this paper, by using the notion of fuzzy points and equality algebras, the notions of fuzzy point equality algebra, equality-subalgebra, and ideal were established. Some characterizations of fuzzy subalgebras were provided by using such concepts. We defined the concepts of (∈, ∈) and (∈, ∈ ∨ q)-fuzzy ideals of equality algebras, discussed some properties, and found some equivalent definitions of them. In addition, we investigated the relation between different kinds of (α, β)-fuzzy subalgebras and (α, β)-fuzzy ideals on equality algebras. Also, by using the notion of (∈, ∈)-fuzzy ideal, we defined two equivalence relations on equality algebras and we introduced an order on classes of X, and we proved that the set of all classes of X by these order is a poset. [ABSTRACT FROM AUTHOR]

This paper focuses on the study and investigation of quotient Riesz algebras, examining their properties and exploring various topologies associated with them. The aim is to gain a better understanding of the behavior and characteristics of these quotient Riesz algebras in relation to their topological structures [ABSTRACT FROM AUTHOR]

We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context - or equivalently for a strongly stratifying ideal - we show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. One of the main consequences is that Han's conjecture holds for an algebra admitting a strongly (co-)stratifying chain whose steps verify Han's conjecture. If Han's conjecture is true for local algebras and an algebra Λ admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for Λ. [ABSTRACT FROM AUTHOR]

Sensor networks are used in an increasing number and variety of application areas, like traffic control or river monitoring. Sensors in these networks measure parameters of interest defined by domain experts and send these measurements to a central location for storage, viewing and analysis. Temporal graph data models, whose nodes contain time-series data reported by the sensors, have been proposed to model and analyze these networks in order to take informed and timely decisions on their operation. Temporal paths are first-class citizens in this model and some classes of them have been identified in the literature. Queries aimed at finding these paths are denoted as (temporal) path queries. In spite of these efforts, many interesting problems remain open and, in this work, we aim at answering some of them. More concretely, we characterize the classes of temporal paths that can be defined in a sensor network in terms of the well-known Allen's temporal algebra. We also show that, out of the 8192 possible interval relations in this algebra, only 11 satisfy two desirable properties that we define: transitivity and robustness. We show how these properties and the paths that satisfy them are relevant in practice by means of a real-world use case consisting of an analysis of salinity that appears close to the Scheldt river in Flanders, Belgium, during high tides occurring in the North Sea. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of single-valued neutrosophic hyper BCKsubalgebras as a generalization and alternative of hyper BCK-algebras and on any given nonempty set constructs at least one single-valued neutrosophic hyper BCK-subalgebra and one a single-valued neutrosophic hyper BCK-ideal. In this study level subsets play the main role in the connection between singlevalued neutrosophic hyper BCK-subalgebras and hyper BCK-subalgebras and the connection between single-valued neutrosophic hyper BCK-ideals and hyper BCK-ideals. The congruence and (strongly) regular equivalence relations are the important tools for connecting hyperstructures and structures, so the major contribution of this study is to apply and introduce a (strongly) regular relation on hyper BCK-algebras and to investigate their categorical properties (quasi commutative diagram) via single-valued neutrosophic hyper BCK-ideals. Indeed, by using the single-valued neutrosophic hyper BCK-ideals, we define a congruence relation on (weak commutative) hyper BCK-algebras that under some conditions is strongly regular and the quotient of any (single-valued neutrosophic)hyper BCK-(sub)algebra via this relation is a (single-valued neutrosophic)(hyper BCKsubalgebra) BCK-(sub)algebra. [ABSTRACT FROM AUTHOR]

In this paper, we shed new light on the Flexible Atom Conjecture. We first give finite representation results for relation algebras 3 3 3 7 , 3 5 3 7 , 7 7 8 3 , 7 8 8 3 , 8 0 8 3 , 8 2 8 3 , 8 3 8 3 , 1 3 1 0 1 3 1 6 , 1 3 1 3 1 3 1 6 , 1 3 1 5 1 3 1 6 and 1 3 1 6 1 3 1 6 . Prior to our paper, only 8 3 8 3 and 1 3 1 6 1 3 1 6 were known to be finitely representable. We accomplish this by generalizing the notion of a relation algebra generated by a Ramsey scheme to the directed (antisymmetric) setting, and then showing that each of these algebras embeds into a finite directed anti-Ramsey scheme. The notion of a directed anti-Ramsey scheme may be of independent interest. We complement our upper bounds with some lower bounds. Namely, we show that any square representation of 3 1 3 7 requires at least 14 points, any square representation of 3 3 3 7 requires at least 11 points, and any square representation of 3 5 3 7 requires at least 12 points. Our technique adapts previous work of Alm et al. [Algebra Univ. (2022)], in that we examine the combinatorial structure induced by the flexible atom. [ABSTRACT FROM AUTHOR]

We equip the type A diagrammatic Hecke category with a special derivation, so that after specialization to characteristic p it becomes a p-dg category. We prove that the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. We conjecture that the p-dg Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztig basis and the p-canonical basis. More precise conjectures will be found in the sequel. Here are some other results contained in this paper. We provide an incomplete proof of the classification of all degree +2 derivations on the diagrammatic Hecke category, and a complete proof of the classification of those derivations for which the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence. We prove that no such derivation exists in simply-laced types outside of finite and affine type A. We also examine a particular Bott-Samelson bimodule in type A_7, which is indecomposable in characteristic 2 but decomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any characteristic, which is the analogue in the p-dg setting of being indecomposable. [ABSTRACT FROM AUTHOR]

In the paper, a basis of identities for the variety generated by the class of groupoids that are generalized subreducts of Tarski's algebra of relations is found. It is also proved that the corresponding class of groupoids does not form a variety. [ABSTRACT FROM AUTHOR]

We compute the matrix elements of SO(3) in any finite-dimensional irreducible representation of sl3. They are expressed in terms of a double sum of products of Krawtchouk and Racah polynomials which generalize the Griffiths–Krawtchouk polynomials. Their recurrence and difference relations are obtained as byproducts of our construction. The proof is based on the decomposition of a general three-dimensional rotation in terms of elementary planar rotations and a transition between two embeddings of sl2 in sl3. The former is related to monovariate Krawtchouk polynomials and the latter, to monovariate Racah polynomials. The appearance of Racah polynomials in this context is algebraically explained by showing that the two sl2 Casimir elements related to the two embeddings of sl2 in sl3 obey the Racah algebra relations. We also show that these two elements generate the centralizer in U(sl3) of the Cartan subalgebra and its complete algebraic description is given. [ABSTRACT FROM AUTHOR]

Two one-parameter families of positively graded Lie superalgebras generated by two elements and two relations that are narrow in the sense of Zelmanov and Shalev are considered. The first family contains the positive part R+ of the Ramond algebra, while the second one contains the positive part NS+ of the Neveu–Schwarz algebra. The results of the article are super analogues of Benoist's theorem on defining the positive part of the Witt algebra by generators and relations. [ABSTRACT FROM AUTHOR]

On a Lie group, we introduce a family of group-invariant metrics and show that the curves invariant under this group are spirals in all the introduced metrics (i.e., they have constant curvatures). An important role is played by an algebra, which we call dual, defined on the same group. The main relation between these algebras is that the trajectories of the one-parameter groups generated by one algebra are invariant curves in the metric that is invariant under the other algebra. The fact that these curves are spirals distinguishes our approach from that of Cartan, who considered the trajectories of one-parameter groups as geodesics in some metric. The presented results are related to the analysis of the geometric meaning of the previously obtained classification of one-dimensional kinetic equations, where invariant curves are the trajectories of particles. [ABSTRACT FROM AUTHOR]

We construct finite-dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We show conversely that every finite-dimensional pointed Hopf algebra over a nonabelian group with nonsimple infinitesimal braiding of rank at least 4 is of this form. We follow the steps of the Lifting Method by Andruskiewitsch-Schneider. Our starting point is the classification of finite-dimensional Nichols algebras over nonabelian groups by Heckenberger-Vendramin, which consist of low-rank exceptions and large-rank families. We prove that the large-rank families are cocycle twists of Nichols algebras constructed by the second author as foldings of Nichols algebras of Cartan type over abelian groups by outer automorphisms. This enables us to give uniform Lie-theoretic descriptions of the large-rank families, prove generation in degree 1, and construct liftings. We also show that every lifting is a cocycle deformation of the corresponding coradically graded Hopf algebra using an explicit presentation by generators and relations of the Nichols algebra. On the level of tensor categories, we construct families of graded extensions of the representation category of a quantum group by a group of diagram automorphism. [ABSTRACT FROM AUTHOR]

We show that the mesh mutations are the minimal relations among the g${\bm{g}}$‐vectors with respect to any initial seed in any finite‐type cluster algebra. We then use this algebraic result to derive geometric properties of the g${\bm{g}}$‐vector fan: we show that the space of all its polytopal realizations is a simplicial cone, and we then observe that this property implies that all its realizations can be described as the intersection of a high‐dimensional positive orthant with well‐chosen affine spaces. This sheds a new light on and extends earlier results of Arkani‐Hamed, Bai, He, and Yan in type A$A$ and of Bazier‐Matte, Chapelier‐Laget, Douville, Mousavand, Thomas, and Yıldırım for acyclic initial seeds. Moreover, we use a similar approach to study the space of polytopal realizations of the g${\bm{g}}$‐vector fans of another generalization of the associahedron: nonkissing complexes (also known as support τ$\tau$‐tilting complexes) of gentle algebras. We show that the space of realizations of the nonkissing fan is simplicial when the gentle bound quiver is brick and 2‐acyclic, and we describe in this case its facet‐defining inequalities in terms of mesh mutations. Along the way, we prove algebraic results on 2‐Calabi–Yau triangulated categories, and on extriangulated categories that are of independent interest. In particular, we prove, in those two setups, an analogue of a result of Auslander on minimal relations for Grothendieck groups of module categories. [ABSTRACT FROM AUTHOR]

In this paper, we push forward the conjecture on a noncommutative version of the maximal ergodic theorem for positive contractions due to Ackoglu. That is, we establish the one‐sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative Lp$L_p$‐spaces for a fixed 1

Given an exterior algebra E over a finite dimension vector space v, and let R = E/I, where I is a graded ideal in E. The relation between the algebra R and Rmon regarding to G-quadratic and LG-quadratic will be investigated. We show that the algebra R is G-quadratic if and only if Rmon is G-quadratic. Furthermore, it has been shown that the algebra R is LG-quadratic if and only if Rmon is LG-quadratic. [ABSTRACT FROM AUTHOR]

Let p be a prime number, Zp the ring of p-adic integers, Qp the field of p-adic numbers. Let K be a complete valued field extension of Qp and LK the valuation ring of K. Let D be a closed unitary subring of the ring LK. Let n ≥ 2 be an integer and let H(2n + 1,D) be the (2n + 1)-dimensional Heisenberg group with entries in D. In [6] we have studied ultrametric Schrödinger representations of the Heisenberg group H(3,D) in two kinds of function spaces. In this paper we extend the study to the (2n+1)-dimensional Heisenberg group with entries in D. We study these representations in the case when the ring D is compact and in the general case their restriction to the algebra of analytic functions on Dn identified with the Tate algebra Tn(K) = K < z1, ..., zn > in n variables. These representations are topologically irreducible. From the last representations one obtains bounded linear operators satisfying Heisenberg commutation relations and theWeyl algebra An(K) as subalgebra of the Banach algebra L(Tn(K)) of bounded linear operators of Tn(K). Its closure fAn(K) in L(Tn(K)) is described and is seen to be a central simple algebra. Notice that this algebra is different from those considered by Pangalos in [11]. [ABSTRACT FROM AUTHOR]

We show how modal quantales arise as convolution algebras Q X of functions from catoids X, multisemigroups equipped with source and target maps, into modal quantales value or weight quantales Q. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in X, Q and Q X . The catoids introduced generalise Schweizer and Sklar's function systems and single-set categories to structures isomorphic to algebras of ternary relations, as they are used for boolean algebras with operators and substructural logics. Our correspondence results support a generic construction of weighted modal quantales from catoids. This construction is illustrated by many examples. We also relate our results to reasoning with stochastic matrices or probabilistic predicate transformers. [ABSTRACT FROM AUTHOR]

This work makes a contribution to the theory of soft sets. It studies the concepts of soft semi-algebras and soft algebras, along with some operations. Then, it examines the relations of soft algebras set to their ordinary (crisp) counterparts. Among other things, we show that every algebra of soft sets induces a collection of ordinary algebras of sets. By using the formulas (in Theorem 7 and Corollary 1), we present a novel construction, allowing us to construct a soft algebra from a system of ordinary algebras of sets. Two examples are presented to show how these formulas can be used in practice. This approach is general enough to be applied to many other (soft) algebraic properties and shows that ordinary algebras contain instruments enabling us to construct soft algebras and to study their properties. As an application, we demonstrate how elements of the generated soft algebra can be used to describe the weather conditions of a region. [ABSTRACT FROM AUTHOR]

Motivated by the pivotal role played by linear operators, many years ago Rota proposed to determine algebraic operator identities satisfied by linear operators on associative algebras, later called Rota's Program on Algebraic Operators. Recent progresses on this program have been achieved in the contexts of operated algebra, rewriting systems and Gröbner-Shirshov bases. These developments also suggest that Rota's insight can be applied to determine operator identities on Lie algebras, and thus to put the various linear operators on Lie algebras in a uniform perspective. This paper carries out this approach, utilizing operated polynomial Lie algebras spanned by nonassociative Lyndon-Shirshov bracketed words. The Lie algebra analog of Rota's program is formulated in terms convergent rewriting systems and equivalently in terms of Gröbner-Shirshov bases. The relation of this Lie algebra analog of Rota's program with Rota's program for associative algebras is established. Applications are given to modified Rota-Baxter operators, differential type operators and Rota-Baxter type operators. [ABSTRACT FROM AUTHOR]

We consider four subsets of the complexified spacetime algebra, namely the real even part, the real odd part, the imaginary even part and the imaginary odd part. This naturally leads to the four connected components of the Lorentz group, supplemented each time by an additional symmetry. We then examine how these four parts impact the Dirac equation and show that four types of matter arise with positive and negative masses as well as positive and negative charges. [ABSTRACT FROM AUTHOR]

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied. [ABSTRACT FROM AUTHOR]

We show that, up to Morita equivalence, any standardly stratified algebra, admits an exact Borel subalgebra. In fact, we show this in the more general case of finite-dimensional algebras possessing an admissible homological system. This generalizes a theorem by Koenig, Külshammer, and Ovsienko, which holds for quasi-hereditary algebras. Our proof follows the same general scheme proposed by these authors, in a more general context: we associate a differential graded tensor algebra with relations, using the structure of A ∞ -algebra of a suitable Yoneda algebra, and use its category of modules to describe the category of filtered modules associated to the given homological system. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new kinds of graphs based on equality algebras. First of all, by using the meet operation we define the notion of zero divisors on equality algebra and study related properties. Then we introduce a meet graph on equality algebra by using zero divisors. In addition, we investigate some conditions for a meet graph to be a complete, connected and a star graph. Also, by using the notion of filters in equality algebras, we define an equivalence relation on equality algebras and then by using the equivalence classes we introduce two kinds of graphs on them. Finally, conditions for a graph based on these classes to be connected, or bipartite, or complete, or planar or an outer-planar graph are provided. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of prime characteristic p. Let kge be a block of a group algebra of a finite group G, with normal defect group P and abelian p' inertial quotient L. Then we show that kge is a matrix algebra over a quantised version of the group algebra of a semidirect product of P with a certain subgroup of L. To do this, we first examine the associated graded algebra, using a Jennings-Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the nonprincipal block in the case of a semidirect product of an extraspecial p-group P of exponent p and order -3 with a quaternion group of order eight with the centre acting trivially. In the case of p = 3, we give explicit generators and relations for the basic algebra as a quantised version of kp. As a second example, we give explicit generators and relations in the case of a group of shape 21+4 : 31+2 in characteristic two. [ABSTRACT FROM AUTHOR]

We extend recent work by Howie, Mathews and Purcell to simplify the calculation of A-polynomials for any family of hyperbolic knots related by twisting. The main result follows from the observation that equations defining the deformation variety that correspond to the twisting are reminiscent of exchange relations in a cluster algebra. We prove two additional results with analogues in the context of cluster algebras: the Laurent phenomenon, and intersection numbers appearing as exponents in the denominator. We demonstrate our results on the twist knots, and on a family of twisted torus knots for which A-polynomials have not previously been calculated. [ABSTRACT FROM AUTHOR]

In this paper, we investigated some related properties and applied the concept of fuzzy congruence relation to a pseudo-UP algebra X. Additionally, we used their level set to define the fuzzy congruence relation on X. The Cartesian product of two fuzzy congruences and the intersection of two fuzzy congruences are discussed. We also provided the connection between the collection of fuzzy pseudo-UP ideals and the collection of fuzzy congruence relations in pseudo-UP algebras. We then demonstrated that there exists an associated algebra X that is a pseudo-UP algebra for each fuzzy ideal ζ and fuzzy congruence ϕζ generated by ζ. Additionally, we created a congruence relation on a pseudo-UP algebra using the fuzzy pseudo-UP ideal and fuzzy congruence relation ϕζ produced by ζ. [ABSTRACT FROM AUTHOR]

Relational databases are a successful model for data storage, and rely on query languages for information retrieval. Most of these query languages are based on relational algebra, a mathematical formalization at the core of relational models. Knowledge graphs are flexible data storage structures that allow for knowledge completion using machine learning techniques. Knowledge hypergraphs generalize knowledge graphs by allowing multi-argument relations. This work studies knowledge hypergraph completion through the lens of relational algebra and its core operations. We explore the space between relational algebra foundations and machine learning techniques for knowledge completion. We investigate whether such methods can capture high-level abstractions in terms of relational algebra operations. We propose a simple embedding-based model called Relational Algebra Embedding (ReAlE) that performs link prediction in knowledge hypergraphs. We show theoretically that ReAlE is fully expressive and can represent the relational algebra operations of renaming, projection, set union, selection, and set difference. We verify experimentally that ReAlE outperforms state-of-the-art models in knowledge hypergraph completion, and in representing each of these primitive relational algebra operations. For the latter experiment, we generate a synthetic knowledge hypergraph, for which we design an algorithm based on the Erdos-Rényi model for generating random graphs. [ABSTRACT FROM AUTHOR]

The algebraic expression 3 + 2 + 6 can be evaluated to 11, but it can also be partially evaluated to 5 + 6. In categorical algebra, such partial evaluations can be defined in terms of the 1-skeleton of the bar construction for algebras of a monad. We show that this partial evaluation relation can be seen as the relation internal to the category of algebras generated by relating a formal expression to its total evaluation. The relation is transitive for many monads which describe commonly encountered algebraic structures, and more generally for BC monads on Set (which are those monads for which the underlying functor and the multiplication are weakly cartesian). We find that this is not true for all monads: we describe a finitary monad on Set for which the partial evaluation relation on the terminal algebra is not transitive. With the perspective of higher algebraic rewriting in mind, we then investigate the compositional structure of the bar construction in all dimensions. We show that for algebras of BC monads, the bar construction has fillers for all directed acyclic configurations in Δn, but generally not all inner horns. [ABSTRACT FROM AUTHOR]

We study the expressive power of the Lara language – a recently proposed unified model for expressing relational and linear algebra operations – both in terms of traditional database query languages and some analytic tasks often performed in machine learning pipelines. Since Lara is parameterized by a set of user-defined functions which allow to transform values in tables, known as extension functions , the exact expressive power of the language depends on how these functions are defined. We start by showing Lara to be expressive complete with respect to a syntactic fragment of relational algebra with aggregation (under the mild assumption that extension functions in Lara can cope with traditional relational algebra operations such as selection and renaming). We then look further into the expressiveness of Lara based on different classes of extension functions, and distinguish two main cases depending on the level of genericity that queries are enforced to satisfy. Under strong genericity assumptions the language cannot express matrix convolution, a very important operation in current machine learning pipelines. This language is also local, and thus cannot express operations such as matrix inverse that exhibit a recursive behavior. For expressing convolution, one can relax the genericity requirement by adding an underlying linear order on the domain. This, however, destroys locality and turns the expressive power of the language much more difficult to understand. In particular, although under complexity assumptions some versions of the resulting language can still not express matrix inverse, a proof of this fact without such assumptions seems challenging to obtain. [ABSTRACT FROM AUTHOR]

In this paper, we first study infinitesimal deformations of a Lie conformal algebra and a Lie conformal algebra with a module (called an LCMod pair), which lead to the notions of the Nijenhuis operator on the Lie conformal algebra and the Nijenhuis structure on the LCMod pair, respectively. Then, by adding compatibility conditions between Nijenhuis structures and O -operators, we introduce the notion of an O N -structure on an LCMod pair and show that an O N -structure gives rise to a hierarchy of pairwise compatible O -operators. In particular, we show that compatible O -operators on a Lie conformal algebra can be characterized by Nijenhuis operators on Lie conformal algebras. Finally, we introduce the notions of the conformal r-matrix-Nijenhuis structure and symplectic-Nijenhuis structure on Lie conformal algebras and study their relations. [ABSTRACT FROM AUTHOR]

In this study, fuzzy ideals in pseudo-hoop algebras are presented. Also, we investigate fuzzy congruences relations on pseudo-hoop algebras induced by fuzzy ideals. By using fuzzy ideals, we create the fuzzy quotient pseudo-hoop algebras and identify and demonstrate the one-to-one relationship between the set of all normal fuzzy ideals of a pseudo-hoop algebra H with conditions p D N and the set of all fuzzy congruences relation on H. Additionally, we investigate the relationship between fuzzy ideals and fuzzy filters in pseudo-hoop algebras by considering the set of complement elements of pseudo-hoop algebras. Lastly, we study fuzzy prime ideals and fuzzy ⊙ -prime ideals and their relation. [ABSTRACT FROM AUTHOR]

In this paper we introduce RL-instances as a way to represent and manipulate gradual conjunctive collections of tuples. Our proposal, based on the theory of Representation by Levels, overcomes the limitations of fuzzy instances in keeping all the properties of conventional Relational Algebra operations. We provide a proof of concept implementation in a conventional relational database management system, where flexible queries can be performed on crisp tables using the same patterns that are used in conventional querying. [ABSTRACT FROM AUTHOR]

Although there have been many fuzzy or probabilistic relational database models proposed for representing and handling imprecise and uncertain information of objects in real-world applications, models combining the relevance and strength of both fuzzy set theory and probability theory appear sporadic. In this paper, we propose a new fuzzy and probabilistic relational database model where the imprecision of an attribute value is represented by a fuzzy set and the uncertainty of a relational tuple is represented by a probability interval. The mass assignment theory is employed to deal with the challenge of integration and computation of both fuzzy sets and probabilities in the same model. The conjunction and disjunction strategies to combine imprecise and uncertain information are introduced. Then the fundamental concepts of the classical relational database model are extended and generalized in this new model. The syntax and semantics of the selection operation are formally defined. Finally, the other important algebraic operations on imprecise attributes and uncertain tuples are developed. [ABSTRACT FROM AUTHOR]

Let 2 < n < ω. Then CAn denotes the class of cylindric algebras of dimension n, RCAn denotes the class of representable CAns, CRCAn denotes the class of completely representable CAns, and NrnCAω(⊆ CAn) denotes the class of n-neat reducts of CAωs. The elementary closure of the class CRCAns (Kn) and the nonelementary class At(NrnCAω) are characterized using two-player zero-sum games, where At is the operator of forming atom structures. It is shown that Kn is not finitely axiomatizable and that it coincides with the class of atomic algebras in the elementary closure of ScNrnCAω where Sc is the operation of forming complete subalgebras. For any class L such that AtNrnCAω ⊆ L ⊆ AtKn, it is proved that SPCmL = RCAn, where Cm is the dual operator to At; that of forming complex algebras. It is also shown that any class K between CRCAn ∩ SdNrnCAω and ScNrnCAn+3 is not first order definable, where Sd is the operation of forming dense subalgebras, and that for any 2 < n < m, any l ≥ n + 3 any any class K such that At(NrnCAm ∩ CRCAn) ⊆ K ⊆ AtScNrnCAl, K is not not first order definable either. [ABSTRACT FROM AUTHOR]

In this paper, we shed new light on the spectrum of the relation algebra we call A n , which is obtained by splitting the non-flexible diversity atom of 6 7 into n symmetric atoms. Precisely, show that the minimum value in Spec (A n) is at most 2 n 6 + o (1) , which is the first polynomial bound and improves upon the previous bound due to Dodd and Hirsch (J Relat Methods Comput Sci 2:18–26, 2013). We also improve the lower bound to 2 n 2 + 4 n + 1 , which is roughly double the trivial bound of n 2 + 2 n + 3 . In the process, we obtain stronger results regarding Spec (A 2) = Spec (32 65) . Namely, we show that 1024 is in the spectrum, and no number smaller than 26 is in the spectrum. Our improved lower bounds were obtained by employing a SAT solver, which suggests that such tools may be more generally useful in obtaining representation results. [ABSTRACT FROM AUTHOR]

We classify n -hereditary monomial algebras in three natural contexts: First, we give a classification of the n -hereditary truncated path algebras. We show that they are exactly the n -representation-finite Nakayama algebras classified by Vaso. Next, we classify partially the n -hereditary quadratic monomial algebras. In the case n = 2 , we prove that there are only two examples, provided that the preprojective algebra is a planar quiver with potential. The first one is a Nakayama algebra and the second one is obtained by mutating k A 3 ⊗ k k A 3 , where A 3 is the Dynkin quiver of type A with bipartite orientation. In the case n ≥ 3 , we show that the only n -representation finite algebras are the n -representation-finite Nakayama algebras with quadratic relations. [ABSTRACT FROM AUTHOR]

We construct a functor that maps C*-correspondences to their Cuntz-Pimsner algebras. Applications include a generalization of the well-known result of Muhly and Solel: Morita equivalent C*-correspondences have Morita equivalent Cuntz-Pimsner algebras; as well as the result of Muhly, Pask, and Tomforde: regular strong shift equivalent C*-correspondences have Morita equivalent Cuntz-Pimsner algebras. [ABSTRACT FROM AUTHOR]

|$\{log\}$| ('setlog') is a satisfiability solver for formulas of the theory of finite sets and finite set relation algebra (FS&RA). As such, it can be used as an automated theorem prover for this theory. |$\{log\}$| is able to automatically prove a number of FS&RA theorems, but not all of them. Nevertheless, we have observed that many theorems that |$\{log\}$| cannot automatically prove can be divided into a few subgoals automatically dischargeable by |$\{log\}$|⁠. The purpose of this work is to present a prototype interactive theorem prover (ITP), called |$\{log\}$| -ITP, providing evidence that a proper integration of |$\{log\}$| into world-class ITP's can deliver a great deal of proof automation concerning FS&RA. An empirical evaluation based on 210 theorems from the TPTP and Coq's SSReflect libraries shows a noticeable reduction in the size and complexity of the proofs with respect to Coq. [ABSTRACT FROM AUTHOR]

In our preceding paper we have introduced the notion of an s -homogeneous triple. In this paper we use this technique to study connected s -homogeneous algebras with two relations. For such algebras, we describe all possible pairs (A , M) , where A is the s -Veronese ring and M is the (s , 1) -Veronese bimodule of the s -homogeneous dual algebra. For each such a pair we classify all the corresponding algebras or give a necessary and sufficient conditions for the algebra under consideration correspond to a given pair. During this process, we have proved that an s -homogeneous algebra with two dimensional s -th component cannot be s -Koszul for s > 2 and discuss the Koszul property for s -homogeneous algebras with two relations. [ABSTRACT FROM AUTHOR]

For a finite group G$G$, we introduce a generalization of norm relations in the group algebra Q[G]$\mathbb {Q}[G]$. We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the arithmetic invariants of the subfields of a normal extension of algebraic number fields with Galois group G$G$. On the algorithmic side, this leads to subfield based algorithms for computing rings of integers, S$S$‐unit groups and class groups. For the S$S$‐unit group computation this yields a polynomial time reduction to the corresponding problem in subfields. We compute class groups of large number fields under GRH, and new unconditional values of class numbers of cyclotomic fields. [ABSTRACT FROM AUTHOR]

Worst-case optimal join algorithms have gained a lot of attention in the database literature. We now count several algorithms that are optimal in the worst case, and many of them have been implemented and validated in practice. However, the implementation of these algorithms often requires an enhanced indexing structure: to achieve optimality one either needs to build completely new indexes or must populate the database with several instantiations of indexes such as B+-trees. Eitherway, this means spending an extra amount of storage space that is typically one or two orders of magnitude more than what is required to store the raw data. We show that worst-case optimal algorithms can be obtained directly from a representation that regards the relations as point sets in variable-dimensional grids, without the need of any significant extra storage. Our representation is a compressed quadtree for the static indexes and a quadtree built on the fly that shares subtrees (which we dub a qdag) for intermediate results.We develop a compositional algorithm to process full join queries under this representation, which simulates navigation of the quadtree of the output, and show that the running time of this algorithm is worst-case optimal in data complexity. We implement our index and compare it experimentally with state-of-the-art alternatives.Our experiments show that our index uses even less space than what is needed to store the data in raw form (and replaces it) and one or two orders of magnitude less space than the other indexes. At the same time, our query algorithm is competitive in time, even sharply outperforming other indexes in various cases. Finally, we extend our framework to evaluate more expressive queries from relational algebra, including not only joins and intersections but also unions and negations. To obtain optimality on those more complex formulas, we introduce a lazy version of qdags we dub lqdags, which allow us navigate over the quadtree representing the output of a formula while only evaluating what is needed from its components.We showthat the running time of our query algorithms on this extended set of operations is worst-case optimal under some constraints. Moving to full relational algebra, we also show that lqdags can handle selections and projections. While worst-case optimality is no longer guaranteed, we introduce a partial materialization scheme that extends results from Deep and Koutris regarding compressed representation of query results. [ABSTRACT FROM AUTHOR]

In noncommutative algebraic geometry an Artin–Schelter regular (AS-regular) algebra is one of the main interests, and every three-dimensional quadratic AS-regular algebra is a geometric algebra, introduced by Mori, whose point scheme is either $\mathbb {P}^{2}$ or a cubic curve in $\mathbb {P}^{2}$ by Artin et al. ['Some algebras associated to automorphisms of elliptic curves', in: The Grothendieck Festschrift, Vol. 1, Progress in Mathematics, 86 (Birkhäuser, Basel, 1990), 33–85]. In the preceding paper by the authors Itaba and Matsuno ['Defining relations of 3-dimensional quadratic AS-regular algebras', Math. J. Okayama Univ. 63 (2021), 61–86], we determined all possible defining relations for these geometric algebras. However, we did not check their AS-regularity. In this paper, by using twisted superpotentials and twists of superpotentials in the Mori–Smith sense, we check the AS-regularity of geometric algebras whose point schemes are not elliptic curves. For geometric algebras whose point schemes are elliptic curves, we give a simple condition for three-dimensional quadratic AS-regular algebras. As an application, we show that every three-dimensional quadratic AS-regular algebra is graded Morita equivalent to a Calabi–Yau AS-regular algebra. [ABSTRACT FROM AUTHOR]

The article is devoted to non-associative algebras with metagroup relations and modules over them. Their functors are studied. Satellites of functors are scrutinized. An exactness of satellite sequences and diagrams is investigated. [ABSTRACT FROM AUTHOR]

This is a companion paper of [7] , where different notions of dimension for triangulated categories are discussed. Here we compute dimensions for some examples of triangulated categories and thus illustrate and motivate material from [7]. Our examples include path algebras of finite ordered quivers, orbifold projective lines, some tensor powers of path algebras in Dynkin quivers of type A and categories, generated by an exceptional pair. [ABSTRACT FROM AUTHOR]

A solution to the problem of low computational efficiency of the relational data model is investigated. An extension of the relational algebra with the help of operations over data sets, i.e., the basic construction of CODASYL DBTG proposals, is suggested. A user is able to choose a method for implementing data links depending on the following requirements for their processing speed: a slow but flexible option based on symbolic addressing typical of relational DBMSs, or a fast but hard option based on direct pointers (relative addressing) typical of DBMSs of the pre-relational era. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the links between congruences, ideals, and homomorphisms in pseudo-UP algebras. Moreover, we establish a theorem which can be viewed as the First Isomorphism Theorem for these algebras. Furthermore, our results may serve as a base to construct other isomorphism theorems for pseudo-UP algebras. [ABSTRACT FROM AUTHOR]

Fix a finite ordinal n ≥ 3 and let α be an arbitrary ordinal. Let CAn denote the class of cylindric algebras of dimension n and RA denote the class of relation algebras. Let PAα(PEAα) stand for the class of polyadic (equality) algebras of dimension α. We reprove that the class CRCAn of completely representable CAns, and the class CRRA of completely representable RAs are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety V between polyadic algebras of dimension n and diagonal free CAns. We show that that the class of completely and strongly representable algebras in V is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class CRRA is not closed under =1,ω. In contrast, we show that given α ≥ ω, and an atomic A 2 PEAα, then for any n < !, NrnA is a completely representable PEAn. We show that for any α ≥ ω, the class of completely representable algebras in certain reducts of PAαs, that happen to be varieties, is elementary. We show that for α ≥ ω, the the class of polyadic-cylindric algebras dimension α, introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension n and PAαs for α an infinite ordinal, proving negative results for the first and positive ones for the second. [ABSTRACT FROM AUTHOR]

In this paper, by using the concept of maximal filter of equality algebra, we introduce radical of equality algebra. Then some equivalence definitions of it and some related properties are investigated. Then by using this notion, we introduce the concept of semi-maximal filter and prime-like filter on equality algebras and the relation between them and other filters of equality algebra are investigated. Finally, by using the notion of prime-like filters, we introduce a topology on equality algebra. [ABSTRACT FROM AUTHOR]

Studying degenerations of moduli spaces of semistable principal bundles on smooth curves leads to the problem of studying moduli spaces on singular curves. In this note, we will see that the moduli spaces of δ-semistable pseudo bundles on a nodal curve become, for large values of δ, the moduli spaces of semistable singular principal bundles. The latter are reasonable candidates for degenerations and a potential basis for further developments as on irreducible nodal curves. In particular, we find a notion of semistability for principal bundles on reducible nodal curves. The understanding of the asymptotic behavior of δ-semistability rests on a lemma from geometric invariant theory. The results allow for the construction of a universal moduli space of semistable singular principal bundles over the moduli space of stable curves. Due to recent work of Wilson, this universal moduli space has a close relation to the sheaf of algebras of conformal blocks. [ABSTRACT FROM AUTHOR]

We develop a multiset query and update language executable in a term rewriting system. Its most remarkable feature, besides non-standard approach to quantification and introduction of fresh values, is non-determinism — a query result is not uniquely determined by the database. We argue that this feature is very useful, e.g., in modelling user choices during simulation or reachability analysis of a data-centric business process — the intended application of our work. Query evaluation is implemented by converting the query into a terminating term rewriting system and normalizing the initial term which encapsulates the current database. A normal form encapsulates a query result. We prove that our language can express any relational algebra query. Finally, we present a simple business process specification framework (and an example specification). Both syntax and semantics of our query language is implemented in Maude. [ABSTRACT FROM AUTHOR]