In [F.-E. Abid and M. Boucetta, Complete description of invariant, associative pseudo-Euclidean metrics on left Leibniz algebras via quadratic Lie algebras, J. Algebra 638 (2024) 358-395; S. Benayadi and S. Hidri, Leibniz algebras with invariant bilinear forms and related Lie algebras, Commun. Algebra 44(8) (2016) 3538-3556] (resp. [S. Benayadi and F. Mhamdi, Invariant bilinear forms on Leibniz superalgebras, J. Algebra Appl. 21(05) (2022) 2250098]) the authors investigated Leibniz algebras (resp. superalgebras) provided with associative, left-invariant, and right-invariant bilinear forms. In this paper, we generalize these types of invariance to Hom-Leibniz algebras, introducing the concepts of α-invariance, α-left-invariance, and α-right-invariance for a Hom-Leibniz algebra (ℒ, ·, α). Focusing on α-R-quadratic Hom-Leibniz algebras, which are (left or right) Hom-Leibniz algebras equipped with symmetric, non-degenerate, and α-right-invariant bilinear forms,we highlight the connections between these Hom-algebras and some other Hom-algebraic structures. More precisely, every α-R-quadratic regular symmetric Hom-Leibniz algebra gives rise to a new type of Hom-algebra, which we call Hom-RS-Lie algebra. We investigate Hom-RS-Lie algebras and provide some interesting information on their structure. Specifically, we characterize firstly α-R-quadratic regular symmetric Hom-Leibniz algebras using the double extension process, which allows us to describe inductively their associated Hom-RS-Lie algebras. Finally, several non-trivial examples of Hom-RS-Lie algebras are included.

Abstract In “Monk Algebras and Ramsey Theory,” J. Log. Algebr. Methods Program. (2022), Kramer and Maddux prove various representability results in furtherance of the goal of finding the smallest weakly representable but not representable relation algebra. They also pose many open problems. In the present paper, we address problems and issues raised by Kramer and Maddux. In particular, we prove that the algebra obtained by splitting the atoms of an n atom Monk algebra is representable for $$n=32$$ n = 32 and $$n=116$$ n = 116 , and hence Proposition 7 in Kramer-Maddux does not generalize. We answer Problem 1(3) in the negative: relation algebra $$1311_{1316}$$ 1311 1316 is not representable. Thus $$1311_{1316}$$ 1311 1316 is a good candidate for the smallest weakly representable but not representable relation algebra.

Abstract The method of extensive full table scanning and unoptimized index construction can only mechanically read data and rigidly process query conditions, resulting in slow query speed. Therefore, the optimization of structured data query speed for the electric energy meter calibration pipeline based on SQL statements is studied. When constructing an efficient query index for electric energy meter calibration data, priority should be given to frequently queried and foreign key column fields. Columns with large cardinality should be selected for indexing to avoid unnecessary or long text columns. Composite indexes should be used reasonably, and the number of fields should be controlled. Useless indexes should be regularly reviewed and cleaned up, and tools should be used to check their usage and pay attention to index failures. Refactoring query logic based on SQL statements, following the rules of relational algebra optimization, prioritizing the execution of highly selective filtering conditions, preprocessing connection operations, converting relevant subqueries into connection operations, eliminating redundant parentheses, adding filtering conditions to aggregate queries, and properly handling operations such as DISTINCT, complex query decomposition materialization. When adjusting the query execution plan, taking Oracle and MySQL databases as examples to understand the execution process, the key lies in the optimizer selection. Oracle focuses on CBO, and multi-table queries pay attention to the table arrangement order. Through analysis operations, adjust the SQL query execution plan for structured data in the electric energy meter calibration pipeline. The experiment shows that the research method has a high index hit rate in various query scenarios, significantly improving the throughput in various differentiated query scenarios. Compared with the comparative method, it exhibits significant query response speed advantages in different data scales and achieves lower and more stable CPU utilization throughout the entire range of data scale changes, effectively improving the response speed of data queries.

Abstract In the present work, we compute quasi-derivations of the Witt algebra and some algebras well related to the Witt algebra. Namely, we prove that each quasi-derivation of the Witt algebra is a sum of a derivation and a $$\frac{1}{2}$$ 1 2 -derivation; a similar result is obtained for the Virasoro algebra. A different situation appears for Lie algebras $$\mathscr {W}(a,b):$$ W ( a , b ) : In the case of $$b=-1,$$ b = - 1 , they do not have interesting examples of quasi-derivations, but the case of $$b\ne -1$$ b ≠ - 1 provides some new non-trivial examples of quasi-derivations. We also completely describe all quasi-derivations of $$\mathscr {W}(a,b).$$ W ( a , b ) . As a corollary, we describe the derivations and quasi-derivations of the Novikov–Witt and admissible Novikov–Witt algebras previously constructed by Bai and his co-authors; and $$\delta $$ δ -derivations and transposed $$\delta $$ δ -Poisson structures on cited Lie algebras. In particular, we proved that each $$\mathscr {W}(a,b)$$ W ( a , b ) admits a non-trivial transposed $$\frac{1}{1-b}$$ 1 1 - b -Poisson structure.

Previous work has axiomatised the cardinality operation in relation algebras, which counts the number of edges of an unweighted graph. We generalise the cardinality axioms to Stone relation algebras, which model weighted graphs, and study the relationships between various axioms for cardinality. This results in simpler cardinality axioms also for relation algebras. We give sufficient conditions for the representability of Stone relation algebras and for Stone relation algebras to be relation algebras. added explanations

In this paper the subject matter is the methods of decomposing conjunctive formulas in the algebra of finite predicates and the possibility of using dependency structures of relational data to build compact logical networks. The work examines the interrelation between operations of relational algebra and predicate algebra, which makes it possible to form a mathematical framework for the rational representation of complex predicate models. The relevance of the study is determined by the need to create high-performance solutions for linguistic information processing and to optimize parallel computational structures. The goal of the work is to develop an effective method of binary predicate decomposition that reduces the number of auxiliary variable values and ensures the transformation of multidimensional predicate formulas into a compact system of binary relations suitable for hardware implementation in the form of logical networks. This approach improves model performance and reduces hardware complexity. To achieve this goal, the following tasks were carried out: analysis of functional, multivalued, and connection dependencies; determination of the conditions for the existence of quantifier-conjunctive decomposition; and investigation of cases in which a predicate can be represented as a system of binary formulas. The limitations of Cartesian decomposition were identified, and it was shown how dependency structures eliminate these drawbacks, forming more compact and structurally justified models. The research methods include the apparatus of finite predicate algebra, the theory of relational dependencies, techniques for constructing auxiliary predicates, variable elimination using existential quantifiers, and modeling processes in the form of logical networks. Special attention is paid to analyzing the structural properties of formulas that influence decomposition efficiency. The results of the research show that the proposed method allows obtaining more compact binary models in comparison to the Cartesian approach. Predicate decomposition applied to a morphological inflection model has confirmed a significant increase in efficiency during hardware implementation of the logical network and improved performance indicators. In the conclusions, it is emphasized that the use of dependency structures is an effective tool for optimizing formal models and opens opportunities for further automation of predicate-structure construction in tasks of logical analysis and knowledge processing.

We show that all Sugihara monoids can be represented as algebras of binary relations, with the monoid operation given by relational composition. Moreover, the binary relations are weakening relations. The first step is to obtain an explicit relational representation of all finite odd Sugihara chains. Our construction mimics that of Maddux (2010), where a relational representation of the finite even Sugihara chains is given. We define the class of representable Sugihara monoids as those which can be represented as reducts of distributive involutive FL-algebras of binary relations. We then show that the class of representable distributive involutive FL-algebras is closed under ultraproducts. This fact is used to demonstrate that the two infinite Sugihara monoids that generate the quasivariety are also representable. From this it follows that all Sugihara monoids are representable. 27 pages, 1 figure

Differential operators map input changes to output changes and form the building blocks of efficient incremental computations. For example, differential operators for relational algebra are used to perform live view maintenance in database systems. However, few differential operators are known and it is unclear how to develop and verify new efficient operators. In particular, we found that differential operators often need to use internal state to selectively cache relevant information, which is not supported by prior work. To this end, we designed a specification for stateful differential operators that allows custom state, yet places sufficient constraints to ensure correctness. We model our specification in Rocq and show that the specification not only guides the design of novel differential operators, but also can capture some of the most sophisticated existing differential operators: database join and Datalog aggregation. We show how to describe complex incremental computations in OCaml by composing stateful differential operators, which we have extracted from Rocq.

Due to the importance of linear algebra and matrix operations in data analytics, there is significant interest in using relational query optimization and processing techniques for evaluating (sparse) linear algebra programs. In particular, in recent years close connections have been established between linear algebra programs and relational algebra that allow transferring optimization techniques of the latter to the former. In this paper, we ask ourselves which linear algebra programs in MATLANG correspond to the free-connex and q-hierarchical fragments of conjunctive first-order logic. Both fragments have desirable query processing properties: free-connex conjunctive queries support constant-delay enumeration after a linear-time preprocessing phase, and q-hierarchical conjunctive queries further allow constant-time updates. By characterizing the corresponding fragments of MATLANG, we hence identify the fragments of linear algebra programs that one can evaluate with constant-delay enumeration after linear-time preprocessing and with constant-time updates. To derive our results, we improve and generalize previous correspondences between MATLANG and relational algebra evaluated over semiring-annotated relations. In addition, we identify properties on semirings that allow to generalize the complexity bounds for free-connex and q-hierarchical conjunctive queries from Boolean annotations to general semirings.

Temporal relational algebras supporting preferences in temporal relational databases: Definition, properties and evaluation

The subject of the study is the methods of intellectual analysis, namely the construction of a decision tree, associative analysis, the identification of patterns between related events based on data presented by a relational model. The purpose of the study is to analyze the features of information units and data structures, using the example of relational systems that affect the technology of knowledge extraction. Tasks: the article solves the following tasks: to consider the relational data model as the most popular and effective data structure used in intelligent information systems for data processing and storage; to analyze the operations of relational algebra, the operational component of the relational data model regarding the application of aggregate functions; to develop a general formal statement of the problem of knowledge extraction from a relational database; to consider the concept of functional associative rules; the ID3 decision tree generation algorithm focused on data processing in relational systems is analyzed. The following methods are implemented: modern view and trends in the field of data mining; features of building information systems based on relational databases, relational algebra, theory of normalization of relations; analysis of literature on the topic of research; comparative analysis. Results achieved: the relational data model is considered as the most effective data structure used in intelligent information systems for data processing and storage. A group of aggregate functions of relational databases is identified and analyzed with respect to key attributes of the relation, which makes it possible to build logical dependencies between information units of the subject area being analyzed. The task of extracting knowledge from the database is formally formulated. The concept of functional associative rules is introduced. The ID3 decision tree generation algorithm focused on data processing in relational systems is carefully analyzed. The semantic network (SN), built on the basis of the proposed approach, allows to increase the efficiency of decision support systems. Conclusions: the universal approach proposed in the article to build a relational data model of an information system for searching for associative patterns in data allows to solve a whole class of typical tasks in which objects are connected by a "many-to-many" relationship or M →N. The relational database model is proposed as a universal information structure for solving associative analysis tasks and presenting knowledge in the form of a semantic network. The examples given in the article confirm the effectiveness of the developed and considered approaches to solving the problem of data mining in the environment of relational systems. Solving the problem of identifying knowledge in data will allow to improve the quality of management decisions made.

This paper presents the development of a software tool that enables the translation of first-order predicate logic with at most three variables into relation algebra. The tool was developed using the Z3 theorem prover, leveraging its capabilities to enhance reliability, generate code, and expedite development. The resulting standalone Python program allows users to translate first-order logic formulas into relation algebra, eliminating the need to work with relation algebra explicitly. This paper outlines the theoretical background of first-order logic, relation algebra, and the translation process. It also describes the implementation details, including validation of the software tool using Z3 for testing correctness. By demonstrating the feasibility of utilizing first-order logic as an alternative language for expressing relation algebra, this tool paves the way for integrating first-order logic into tools traditionally relying on relation algebra as input. 15 pages, 2 figures, 2 tables, to be published in Fundamenta Informaticae

In this paper, we consider relational structures arising from Comer's finite field construction, where the cosets need not be sum free. These Comer schemes generalize the notion of a Ramsey scheme and may be of independent interest. As an application, we give the first finite representation of $34_{65}$. This leaves $33_{65}$ as the only remaining relation algebra in the family $N_{65}$ with a flexible atom that is not known to be finitely representable. Motivated by this, we complement our upper bounds with some lower bounds. Using a SAT solver, we show that $33_{65}$ is not finitely representable on fewer than $24$ points, and that $33_{65}$ does not admit a cyclic group representation on fewer than $120$ points. We also employ a SAT solver to show that $34_{65}$ is not representable on fewer than $24$ points. Fundamenta Informaticae final journal version; previous conference version appeared in RAMiCS 2023

We introduce the general notions of an index and a core of a relation. We postulate a limited form of the axiom of choice -- specifically that all partial equivalence relations have an index -- and explore the consequences of adding the axiom to standard axiom systems for point-free reasoning. Examples of the theorems we prove are that a core/index of a difunction is a bijection, and that the so-called ``all or nothing'' axiom used to facilitate pointwise reasoning is derivable from our axiom of choice. Fundamenta Informaticae, Volume 195, Issue 1-4, Article 2, 2025

In this paper, we show that the problem of deciding for a given finite relation algebra [Formula: see text] whether the network satisfaction problem for [Formula: see text] can be solved by the k-consistency procedure, for some [Formula: see text], is undecidable. For the important class of finite relation algebras [Formula: see text] with a normal representation, however, the decidability of this problem remains open. We show that if [Formula: see text] is symmetric and has a flexible atom, then the question whether [Formula: see text] can be solved by k-consistency, for some [Formula: see text], is decidable (even in polynomial time in the number of atoms of [Formula: see text]). This result follows from a more general sufficient condition for the correctness of the k-consistency procedure for finite symmetric relation algebras. In our proof we make use of a result of Alexandr Kazda about finite binary conservative structures.

Abstract Character tables of finite groups and closely related commutative algebras have been investigated recently using new perspectives arising from the AdS/CFT correspondence and low-dimensional topological quantum field theories. Two important elements in these new perspectives are physically motivated definitions of quantum complexity for the algebras and a notion of row-column duality. These elements are encoded in properties of the character table of a group G and the associated algebras, notably the centre of the group algebra and the fusion algebra of irreducible representations of the group. Motivated by these developments, we define a notion of generator complexity for commutative Frobenius algebras with combinatorial bases. In the context of finite groups, this gives rise to row and column versions of generator complexity for character tables. We investigate the relation between these complexities under the exchange of rows and columns. We observe regularities that arise in the statistical averages over small character tables and propose corresponding conjectures for arbitrarily large character tables.

DNAQL is a query language for databases implemented as DNA molecules in solution, with well-known DNA wetlab procedures as operators. DNAQL focusses on faithful implementability in DNA, and, as a result, does not include powerful, but difficult to implement, features like nonterminating/recursive hybridization needed for general computation (i.e., Turing-completeness). In this paper we show that DNAQL is relationally complete, i.e., the full relational algebra can be performed by DNAQL programs.

Codd's Theorem, a fundamental result of database theory, asserts that relational algebra and relational calculus have the same expressive power on relational databases. We explore Codd's Theorem for databases over semirings and establish two different versions of this result for such databases: the first version involves the five basic operations of relational algebra, while in the second version the division operation is added to the five basic operations of relational algebra. In both versions, the difference operation of relations is given semantics using semirings with monus, while on the side of relational calculus a limited form of negation is used. The reason for considering these two different versions of Codd's theorem is that, unlike the case of ordinary relational databases, the division operation need not be expressible in terms of the five basic operations of relational algebra for databases over an arbitrary positive semiring; in fact, we show that this inexpressibility result holds for bag databases, as well as for databases over the tropical semiring.

We associate to a quiver and a subquiver .Q; F / a stopped Weinstein manifold X whose Legendrian attaching link is a singular Legendrian unknot link .We prove that the relative Ginzburg algebra of .Q; F / is quasi-isomorphic to the Chekanov-Eliashberg dg-algebra of .It follows that the Chekanov-Eliashberg dg-algebra of relative to its boundary dg-subalgebra, and the Orlov functor associated to the partially wrapped Fukaya category of X both admit a strong relative smooth Calabi-Yau structure.53D35, 53D42; 16E45 1. Introduction 3737 2. Relative Ginzburg algebras 3741 3. Stopped Weinstein manifold associated to a quiver and a frozen subquiver 3742 4. Proof of the main theorem 3747

Abstract We present a special and attractive basis for the exceptional Lie algebra G 2 , which turns G 2 into a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> -graded Lie algebra. There are two basis elements for each degree of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:msubsup> <mml:mo>∖</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> </mml:math> , thus yielding 14 basis elements. We give a general and simple closed form expression for commutators between these basis elements. Next, we use this <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> -grading in order to examine graded color algebras. Our analysis yields three different <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> -graded color algebras of type G 2 . Since the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> -grading is not compatible with a Cartan–Weyl basis of G 2 , we also study another grading of G 2 . This is a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> -grading, compatible with a Cartan–Weyl basis, and for which we can also construct a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> -graded color algebra of type G 2 .