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In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of α -abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we study crossed modules of Hom-Lie algebras showing their equivalence with cat1-Hom-Lie algebras, and we introduce α -crossed modules to have a better understanding of the third cohomology group. [ABSTRACT FROM AUTHOR]

This paper studies the complexity of query evaluation for databases whose relations are partially ordered; the problem commonly arises when combining or transforming ordered data from multiple sources. We focus on queries in a useful fragment of SQL, namely positive relational algebra with aggregates, whose bag semantics we extend to the partially ordered setting. Our semantics leads to the study of two main computational problems: the possibility and certainty of query answers. We show that these problems are respectively NP-complete and coNP-complete, but identify tractable cases depending on the query operators or input partial orders. We further introduce a duplicate elimination operator and study its effect on the complexity results. [ABSTRACT FROM AUTHOR]

Theoretical models used in database research often have subtle differences with those occurring in practice. One particular mismatch that is usually neglected concerns the use of marked nulls to represent missing values in theoretical models of incompleteness, while in an SQL database these are all denoted by the same syntactic object. It is commonly argued that results obtained in the model with marked nulls carry over to SQL, because SQL nulls can be interpreted as Codd nulls , which are simply marked nulls that do not repeat. This argument, however, does not take into account that even simple queries may produce answers where distinct occurrences of do in fact denote the same unknown value. For such queries, interpreting SQL nulls as Codd nulls would incorrectly change the semantics of query answers. To use results about Codd nulls for real-life SQL queries, we need to understand which queries preserve the Codd interpretation of SQL nulls. We show, however, that the class of relational algebra queries preserving Codd interpretation is not recursively enumerable, which necessitates looking for sufficient conditions for such preservation. Those can be obtained by exploiting the information provided by NOT NULL constraints on the database schema. We devise mild syntactic restrictions on queries that guarantee preservation, do not limit the full expressiveness of queries on databases without nulls, and can be checked efficiently. • SQL nulls are commonly interpreted as non-repeating marked nulls, but even simple queries may produce answers that break this interpretation. • The class of queries preserving the Codd interpretation of SQL nulls cannot be captured syntactically. • Sufficient syntactic restrictions for preservation can be obtained by leveraging NOT NULL constraints on the database schema. [ABSTRACT FROM AUTHOR]

Every Lie algebra over a field E gives rise to new Lie algebras over any subfield F ⊆ E by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of the original Lie algebra, in particular the question how much of the original Lie algebra can be recovered from its underlying Lie algebra over subfields F. By introducing the conjugate of a Lie algebra we show that in some specific cases the Lie algebra is completely determined by its underlying Lie algebra. Furthermore we construct examples showing that these assumptions are necessary. As an application, we give for every positive n an example of a real 2-step nilpotent Lie algebra which has exactly n different bi-invariant complex structures. This answers an open question by Di Scala, Lauret and Vezzoni motivated by their work on quasi-Kähler Chern-flat manifolds in differential geometry. The methods we develop work for general Lie algebras and for general Galois extensions F ⊆ E , in contrast to the original question which only considered nilpotent Lie algebras of nilpotency class 2 and the field extension R ⊆ C. We demonstrate this increased generality by characterizing the complex Lie algebras of dimension ≤4 which are defined over R and over Q. [ABSTRACT FROM AUTHOR]

A proper (simple) relation algebra is a collection A of binary relations on a set U such that the top element of the algebra is U × U , and A is closed under union, intersection, complementation, composition, converse and contains the identity relation on U. Proper relation algebras can be constructed using U = Z / p Z as a base set using a method due to Comer. The cycle structure of such an algebra must, in general, be determined a posteriori , normally with the aid of a computer. In this paper, we give an improved algorithm for checking the cycle structure that reduces the time complexity from O (p 2) to O (p). [ABSTRACT FROM AUTHOR]

In this paper, we consider relations of the Lie algebra of holomorphic vector fields and Hodge numbers of a compact pseudo-Kähler nilmanifold. [ABSTRACT FROM AUTHOR]

We demonstrate that commutativity of numerous one-dimensional subalgebras in W 1 + ∞ algebra, i.e. the existence of many non-trivial integrable systems described in recent arXiv:2303.05273 follows from the subset of relations in algebra known as Serre relations. No other relations are needed for commutativity. The Serre relations survive the deformation to the affine Yangian Y ( gl ˆ 1) , hence the commutative subalgebras do as well. A special case of the Yangian parameters corresponds to the β -deformation. The preservation of Serre relations can be thought of a selection rule for proper systems of commuting β -deformed Hamiltonians. On the contrary, commutativity in the extended family associated with "rational (non-integer) rays" is not reduced to the Serre relations, and uses also other relations in the W 1 + ∞ algebra. Thus their β -deformation is less straightforward. [ABSTRACT FROM AUTHOR]

Not all planar algebras can encode the algebraic structure of a Yang–Baxter integrable model described in terms of a so-called homogeneous transfer operator. In the family of subfactor planar algebras, we focus on the ones known as singly generated and find that the only such planar algebras underlying homogeneous Yang–Baxter integrable models are the so-called Yang–Baxter relation planar algebras. According to a result of Liu, there are three such planar algebras: the well-known Fuss–Catalan and Birman–Wenzl–Murakami planar algebras, in addition to one more which we refer to as the Liu planar algebra. The Fuss–Catalan and Birman–Wenzl–Murakami algebras are known to underlie Yang–Baxter integrable models, and we show that the Liu algebra likewise admits a Baxterisation. We also show that the homogeneous transfer operator describing a model underlied by a singly generated Yang–Baxter relation planar algebra is polynomialisable, meaning that it is polynomial in a spectral-parameter-independent element of the algebra. [ABSTRACT FROM AUTHOR]

Ordinary differential equations (ODEs) are widespread in many natural sciences including chemistry, ecology, and systems biology, and in disciplines such as control theory and electrical engineering. Building on the celebrated molecules-as-processes paradigm, they have become increasingly popular in computer science, with high-level languages and formal methods such as Petri nets, process algebra, and rule-based systems that are interpreted as ODEs. We consider the problem of comparing and minimizing ODEs automatically. Influenced by traditional approaches in the theory of programming, we propose differential equivalence relations. We study them for a basic intermediate language, for which we have decidability results, that can be targeted by a class of high-level specifications. An ODE implicitly represents an uncountable state space, hence reasoning techniques cannot be borrowed from established domains such as probabilistic programs with finite-state Markov chain semantics. We provide novel symbolic procedures to check an equivalence and compute the largest one via partition refinement algorithms that use satisfiability modulo theories. We illustrate the generality of our framework by showing that differential equivalences include (i) well-known notions for the minimization of continuous-time Markov chains (lumpability), (ii) bisimulations for chemical reaction networks recently proposed by Cardelli et al., and (iii) behavioral relations for process algebra with ODE semantics. Using ERODE, the tool that implements our techniques, we are able to detect equivalences in biochemical models from the literature that cannot be reduced using competing automatic techniques. [ABSTRACT FROM AUTHOR]

Abstract Binary Constraint Problems have traditionally been considered as Network Satisfaction Problems over some relation algebra. A constraint network is satisfiable if its nodes can be mapped into some representation of the relation algebra in such a way that the constraints are preserved. A qualitative representation ϕ is like an ordinary representation, but instead of requiring that (a ; b) ϕ is the composition a ϕ ∘ b ϕ of the relations a ϕ and b ϕ , as we do for ordinary representations, we only require that c ϕ ⊇ a ϕ ∘ b ϕ ⇔ c ≥ a ; b , for each c in the algebra. A constraint network is qualitatively satisfiable if its nodes can be mapped to elements of a qualitative representation, preserving the constraints. If a constraint network is satisfiable then it is clearly qualitatively satisfiable, but the converse can fail, as we show. However, for a wide range of relation algebras including the point algebra, the Allen Interval Algebra, RCC8 and many others, a network is satisfiable if and only if it is qualitatively satisfiable. Unlike ordinary composition, the weak composition arising from qualitative representations need not be associative, so we can generalise by considering network satisfaction problems over non-associative algebras. We prove that computationally, qualitative representations have many advantages over ordinary representations: whereas many finite relation algebras have only infinite representations, every finite qualitatively representable algebra has a finite qualitative representation; the representability problem for (the atom structures of) finite non-associative algebras is NP-complete ; the network satisfaction problem over a finite qualitatively representable algebra is always in NP ; the validity of equations over qualitative representations is co-NP-complete. On the other hand we prove that there is no finite axiomatisation of the class of qualitatively representable algebras. [ABSTRACT FROM AUTHOR]

Abstract In Relational Algebra, the operator Division (÷) is an intuitive tool used to write queries with the concept of "for all", and thus, it is constantly required in real applications. However, as we demonstrate here, the division does not support many of the needs common to modern applications, particularly those that involve complex data analysis, such as processing images, audio, genetic data, large graphs, fingerprints, and many other "non-traditional" data types. The main issue is the existence of intrinsic comparisons of attribute values in the operator, which, by definition, are always performed by identity (=), despite the fact that complex data must be compared by similarity. Recent works focus on supporting similarity comparison in relational operators, but no one treats the division. This paper presents the new Similarity-aware Division (÷ ˆ) operator. Our novel operator is naturally well suited to answer queries with an idea of "candidate elements and exigencies" to be performed on complex data from modern applications. For example, it is potentially useful to support agriculture, genetic analyses, digital library search, prospective client identification, and even to help controlling the quality of manufactured products in industry. We validate our proposals by studying the first two of these applications. Highlights • The relational division database operator is not suited for complex data analysis. • A novel similarity-aware division operator is proposed to process complex data. • The similarity-aware division is formally defined. • Fast and scalable algorithms for the similarity-aware division are presented. • The new division operator is validated in case studies in Agriculture and Genetics. [ABSTRACT FROM AUTHOR]

Abstract A relation algebra is called measurable when its identity is the sum of measurable atoms, where an atom is called measurable if its square is the sum of functional elements. In this paper we show that atomic measurable relation algebras have rather strong structural properties: they are constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. An atomic and complete measurable relation algebra is completely representable if and only if there is a stronger coordination between these isomorphisms induced by a scaffold (the shifting cosets are not needed in this case). We also prove that a measurable relation algebra in which the associated groups are all finite is atomic. [ABSTRACT FROM AUTHOR]

In this paper we consider, from a computational point of view, the problem of classifying logics within the Leibniz and Frege hierarchies typical of abstract algebraic logic. The main result states that, for logics presented syntactically, this problem is in general undecidable. More precisely, we show that there is no algorithm that classifies the logic of a finite consistent Hilbert calculus in the Leibniz and in the Frege hierarchies. [ABSTRACT FROM AUTHOR]

In this paper, we prove that relation-extensions of quasi-tilted algebras are 2-Calabi–Yau tilted. With the objective of describing the module category of a cluster-tilted algebra of euclidean type, we define the notion of reflection so that any two local slices can be reached one from the other by a sequence of reflections and coreflections. We then give an algorithmic procedure for constructing the tubes of a cluster-tilted algebra of euclidean type. Our main result characterizes quasi-tilted algebras whose relation-extensions are cluster-tilted of euclidean type. [ABSTRACT FROM AUTHOR]

We describe, in an algebraic way, the κ -deformed extended Snyder models, that depend on three parameters β , κ and λ , which in a suitable algebra basis are described by the de Sitter algebras o (1 , N). The commutation relations of the algebra contain a parameter λ , which is used for the calculations of perturbative expansions. For such κ -deformed extended Snyder models we consider the Heisenberg double with dual generalized momenta sector, and provide the respective generalized quantum phase space depending on three parameters mentioned above. Further, we study for these models an alternative Heisenberg double, with the algebra of functions on de Sitter group. In both cases we calculate the formulae for the cross commutation relations between generalized coordinate and momenta sectors, at linear order in λ. We demonstrate that in the commutators of quantum space-time coordinates and momenta of the quantum-deformed Heisenberg algebra the terms generated by κ -deformation are dominating over β -dependent ones for small values of λ. [ABSTRACT FROM AUTHOR]

For any two complex numbers a and b , V i r ( a , b ) is a central extension of W ( a , b ) which is universal in the case ( a , b ) ≠ ( 0 , 1 ) , where W ( a , b ) is the Lie algebra with basis { L n , W n | n ∈ Z } and relations [ L m , L n ] = ( n − m ) L m + n , [ L m , W n ] = ( a + n + b m ) W m + n , [ W m , W n ] = 0 . In this paper, we construct and classify a class of non-weight modules over the algebra V i r ( a , b ) which are free U ( C L 0 ⊕ C W 0 ) -modules of rank 1. It is proved that such modules can only exist for a ∈ Z . [ABSTRACT FROM AUTHOR]

In this paper, we study the incidence algebra T of the attenuated space poset A q ( N , M ) . We consider the following topics. We consider some generators of T : the raising matrix R , the lowering matrix L , and a certain diagonal matrix K . We describe some relations among R , L , K . We put these relations in an attractive form using a certain matrix S in T . We characterize the center Z ( T ) . Using Z ( T ) , we relate T to the quantum group U τ ( sl 2 ) with τ 2 = q . We consider two elements A , A ⁎ in T of a certain form. We find necessary and sufficient conditions for A , A ⁎ to satisfy the tridiagonal relations. Let W denote an irreducible T -module. We find necessary and sufficient conditions for the above A , A ⁎ to act on W as a Leonard pair. [ABSTRACT FROM AUTHOR]

Let B be the split extension of a finite dimensional algebra C by a C - C -bimodule E . We define a morphism of associative graded algebras φ ⁎ : HH ⁎ ( B ) → HH ⁎ ( C ) from the Hochschild cohomology of B to that of C , extending similar constructions for the first cohomology groups made and studied by Assem, Bustamante, Igusa, Redondo and Schiffler. In the case of a trivial extension B = C ⋉ E , we give necessary and sufficient conditions for each φ n to be surjective. We prove the surjectivity of φ 1 for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of φ 1 for any trivial extension, and give a more precise description of this kernel in the case of relation extensions. [ABSTRACT FROM AUTHOR]

In this paper we report about an approach to establish safety properties of cooperating systems in polynomial time, the state space of which can be exponentially large in the number of cooperating subsystems. It consists of constructing a family of so-called inducers of a system and the reduction of these by an operator we call Edge-Match. Furthermore, we draw a connection between our approach and the theory of relational databases. Aside from pointing out an interesting connection between these fields we use this connection to apply results from the theory of relational databases to our approach. [ABSTRACT FROM AUTHOR]

Constraint Databases represent complex data by means of formulas described by constraints (equations, inequations or Boolean combinations of both). Commercial database management systems allow the storage and efficient retrieval of classic data, but for complex data a made-to-measure solution combined with expert systems for each type of problem are necessary. Therefore, in the same way as commercial solutions of relational databases permit storing and querying classic data, we propose an extension of the Selection Operator for complex data stored, and an extension of SQL language for the case where both classic and constraint data need to be managed. This extension shields the user from unnecessary details on how the information is stored and how the queries are evaluated, thereby enlarging the capacity of expressiveness for any commercial database management system. In order to minimize the selection time, a set of strategies have been proposed, which combine the advantages of relational algebra and constraint data representation. [ABSTRACT FROM AUTHOR]

Motivated by results of Aujla and Silva [3], we give several more precise weak majorization and eigenvalue inequalities for some matrix versions of the famous Jensen inequality with regard to a convexity. Our main results are then applied to log convex functions. As an application, we obtain refinements of some well-known matrix inequalities. [ABSTRACT FROM AUTHOR]

While all relational database systems are based on the bag data model, much of theoretical research still views relations as sets. Recent attempts to provide theoretical foundations for modern data management problems under the bag semantics concentrated on applications that need to deal with incomplete relations, i.e., relations populated by constants and nulls. Our goal is to provide a complete characterization of the complexity of query answering over such relations in fragments of bag relational algebra. The main challenges that we face are twofold. First, bag relational algebra has more operations than its set analog (e.g., additive union, max-union, min-intersection, duplicate elimination) and the relationship between various fragments is not fully known. Thus we first fill this gap. Second, we look at query answering over incomplete data, which again is more complex than in the set case: rather than certainty and possibility of answers, we now have numerical information about occurrences of tuples. We then fully classify the complexity of finding this information in all the fragments of bag relational algebra. • Database systems are based on bags, but theoretical research focuses on sets • Classification of relative expressive power of bag relational algebra fragments • Complexity of computing certain and possible answers in those fragments. [ABSTRACT FROM AUTHOR]

Abstract: A Catalan pair is a pair of binary relations satisfying some axioms. These pairs are enumerated by the well-known Catalan numbers, and have been introduced in Disanto et al. (2010) [2] with the aim of giving a common language to many structures counted by Catalan numbers. Here, a simple method is given to pass from the recursive definition of a generic Catalan structure to the recursive definition of the Catalan pair on the same structure, thus giving an automatic way of interpreting Catalan structures in terms of Catalan pairs. Our method is applied to several well-known Catalan structures, focusing on the combinatorial meaning of the relations and in each case considered. [Copyright &y& Elsevier]

Modern data-intensive applications have to manage huge quantities of streaming/relational data and need advanced query capabilities involving combinations of continuous queries (CQs) and one-time queries (OTQs) also requiring the verification of complex temporal conditions. In this paper, we go beyond the disjointed panorama of current approaches and adopt a new holistic approach to the integration of stream processing capabilities into the temporal database world based on the streaming table concept. To this end, we propose a full-fledged query interface composed of a TSQL2-like query language with an underlying algebraic framework. The algebraic framework, which is aimed at implementing the query interface on top of a working DBMS, is made up of: (a) the extended temporal algebra TA ⋆ supporting OTQs with an hybrid temporal semantics (sequenced and non-sequenced); (b) the continuous temporal algebra CTA that extends TA ⋆ with window expressions for CQ specification; (c) the translation of CTA expressions into TA ⋆ ones that can be executed by a traditional DBMS with an extended kernel. • Unification of streaming, temporal, and relational querying for advanced analytics. • Full acknowledgment of the inherently temporal nature of streaming data. • Definition of the Continuous Temporal Algebra. • Unified semantics for one-time and continuous queries. • Implementability on standard DBMS technology to leverage on well-founded procedures. [ABSTRACT FROM AUTHOR]

The industry-wide adoption of graph databases has been hindered due to the fragmentation in syntax and semantics of available graph query languages. As a result, several projects have been proposed by industry and academia to develop a standard query language by integrating features from existing practical graph query languages. A significant factor that can impact query language integration is the lack of common theoretical language formalisms. We propose common formalisms by extending conjunctive queries and union of conjunctive queries with Tarski's relation algebra ( CQT / UCQT ). We use common graph query patterns to compare the expressive power of ( CQT / UCQT ) with two practical graph query languages - C y p h e r and P G Q L. The query languages are analysed on the core features of graph pattern matching and graph navigation, revealing the common and exclusive characteristics for these languages. Overall, our study serves as a formal basis for comparing existing graph query languages and assists the move towards query language integration and interoperability between available graph database technologies. • There is a lack of explicit mapping between practical graph query languages and theoretical language formalisms. • Current formalisms used by practical graph query languages are not expressive enough. • Extension of conjunctive queries and union of conjunctive queries with Tarski's relation algebra. • Use of extended formalisms to compare the expressiveness of practical graph query languages. • A comprehensive comparison of two practical graph query languages Cypher and PGQL. [ABSTRACT FROM AUTHOR]

Abstract: In social choice theory the Banks set is a well-established choice set for tournaments that consists of the undominated elements of the maximal subtournaments. For non-complete dominance relations J. Duggan proposed three possibilities to modify it. We develop relation-algebraic specifications to compute the Banks set, Duggan’s modifications, and variants of them. All these specifications are algorithmic and can directly be translated into the programming language of the computer algebra system RelView. We show that the system is well suited for computing and visualizing the Banks set, its modifications, and the objects to be associated with them. [Copyright &y& Elsevier]

Abstract: In this paper we classify algebraic Anosov actions of nilpotent Lie groups on closed manifolds, extending the previous results by P. Tomter (1970, 1975) [17,18]. We show that they are all nil-suspensions over either suspensions of Anosov actions of on nilmanifolds, or (modified) Weyl chamber actions. We check the validity of the generalized Verjovsky conjecture in this algebraic context. We also point out an intimate relation between algebraic Anosov actions and Cartan subalgebras in general real Lie groups. [Copyright &y& Elsevier]

Abstract: The single defining relation of the algebra of -invariants of triples of 3×3 matrices is explicitly computed. Connections to some other prominent algebras of invariants are pointed out. [Copyright &y& Elsevier]

Abstract: We show that for Multiplicative Exponential Linear Logic (without weakenings) the syntactical equivalence relation on proofs induced by cut-elimination coincides with the semantic equivalence relation on proofs induced by the multiset based relational model: one says that the interpretation in the model (or the semantics) is injective. We actually prove a stronger result: two cut-free proofs of the full multiplicative and exponential fragment of linear logic whose interpretations coincide in the multiset based relational model are the same “up to the connections between the doors of exponential boxes”. [Copyright &y& Elsevier]

Abstract: We describe a simple computing technique for solving independence and domination problems on rectangular chessboards. It rests upon relational modeling and uses the BDD-based specific purpose computer algebra system RelView for the evaluation of the relation-algebraic expressions that specify the problems’ solutions and the visualization of the computed results. The technique described in the paper is very flexible and especially appropriate for experimentation. It can easily be applied to other chessboard problems. [Copyright &y& Elsevier]

Abstract: Problem statements often resort to superlatives such as in e.g. “… the smallest such number”, “… the best approximation”, “… the longest such list” which lead to specifications made of two parts: one defining a broad class of solutions (the easy part) and the other requesting one particular such solution, optimal in some sense (the hard part). This article introduces a binary relational combinator which mirrors this linguistic structure and exploits its potential for calculating programs by optimization. This applies in particular to specifications written in the form of Galois connections, in which one of the adjoints delivers the optimal solution. The framework encompasses re-factoring of results previously developed by Bird and de Moor for greedy and dynamic programming, in a way which makes them less technically involved and therefore easier to understand and play with. [Copyright &y& Elsevier]

Abstract: That matrices of relations also obey the rules of relation algebra is well known. When the powerset ordering is considered, partialities may be conceived as lattice-continuous mappings — corresponding to existential images which are often studied independently. A partiality is suited to describe progress of yet partial information or availability. This has already been presented in Schmidt (2006) . Matrices of partialities will considerably improve the possibility to study non-strictness, streams, partial evaluation, and net properties in a compact relation-algebraic form. They seem, however, to lead inevitably to some borderline cases as the Boolean lattice 0 and row-less matrices. It will be shown how these can be fruitfully applied concerning constructions with temporarily non-connected relation algebras. [Copyright &y& Elsevier]

Abstract: In this paper, we first describe a homogenization methodology with the aim of establishing strain gradient constitutive relations for heterogeneous materials. The methodology presented in this work includes two main steps. The first one is the construction of the average strain-energy density for a well-chosen RVE by using a homogenization technique. The second one is the transformation of the obtained average strain-energy density to that for the continuum. An important characteristic of this method is its self-consistency with respect to the choice of the RVE: the strain gradient constitutive law built by using the present method is independent of the size and the form of the RVE. In the frame of this homogenization procedure, we have constructed a strain gradient constitutive relation for a two-dimensional elastic material with many microcracks by adopting the self-consistent scheme. It was shown that the effective behavior of cracked solids depends not only on the crack density but also on the average crack size with which the strain gradient is associated. The proposed constitutive relation provides a starting point for the development of an evolution law of damage including strain gradient effect, which will be presented in the second part of this work. [Copyright &y& Elsevier]

Abstract: We formulate two open problems related to and, in a sense, suggested by the Reiterman–Tholen characterization of effective descent morphisms of topological spaces. [Copyright &y& Elsevier]

Abstract: We study the Boolean algebras of regular languages, context-sensitive languages and decidable languages, respectively, over any alphabet. It is well known that , with proper inclusions. After observing that these Boolean algebras are all isomorphic, we study some immunity properties: for instance we prove that for every coinfinite decidable language there exists a decidable language such that , is infinite, and there is no context-sensitive language , with unless is finite; similarly, for every coinfinite regular language there exists a context-sensitive language such that , is infinite, and there is no regular language such that , unless is finite. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the concept of -intuitionistic fuzzy sub-hemiring, (left, right, two sided) ideals of hemirings, where are any two of with , by using belongs to relation and quasi-coincidence with relation between intuitionistic fuzzy points and intuitionistic fuzzy sets, and investigate related properties. Moreover, we define prime(semiprime) -intuitionistic fuzzy ideals of hemirings and investigate some different properties of these ideals. [Copyright &y& Elsevier]

Abstract: The use of linguistic variables and fuzzy propositions in the interval-valued -Fuzzy contexts can be an interesting tool to extract a more complete information from them. In this paper, we analyze three different situations. First, we obtain significant relations in order to study all the objects and attributes of the interval-valued -Fuzzy context by means of the interval-valued -Fuzzy concepts. After that, we show how to replace the erroneous values to be able to study in a suitable way the context. Finally, we use the linguistic labels to obtain a subcontext that represents our interest of study. We also show an experimental evaluation in the paper. [Copyright &y& Elsevier]

Abstract: More than 15 years ago, a set of qualitative spatial relations between oriented straight line segments (dipoles) was suggested by Schlieder. However, it turned out to be difficult to establish a sound constraint calculus based on these relations. In this paper, we present the results of a new investigation into dipole constraint calculi which uses algebraic methods to derive sound results on the composition of relations of dipole calculi. This new method, which we call condensed semantics, is based on an abstract symbolic model of a specific fragment of our domain. It is based on the fact that qualitative dipole relations are invariant under orientation preserving affine transformations. The dipole calculi allow for a straightforward representation of prototypical reasoning tasks for spatial agents. As an example, we show how to generate survey knowledge from local observations in a street network. The example illustrates the fast constraint-based reasoning capabilities of dipole calculi. We integrate our results into two reasoning tools which are publicly available. [Copyright &y& Elsevier]

Abstract: Coalgebras can be seen as a natural abstraction of Kripke frames. In the same sense, coalgebraic logics are generalised modal logics. In this paper, we give an overview of the basic tools, techniques and results that connect coalgebras and modal logic. We argue that coalgebras unify the semantics of a large range of different modal logics (such as probabilistic, graded, relational, conditional) and discuss unifying approaches to reasoning at this level of generality. We review languages defined in terms of the so-called cover modality, languages induced by predicate liftings as well as their common categorical abstraction, and present (abstract) results on completeness, expressiveness and complexity in these settings, both for basic languages as well as a number of extensions, such as hybrid languages and fixpoints. [Copyright &y& Elsevier]

Abstract: It is known that a class of graphs defined by a single forbidden induced subgraph is well-quasi-ordered by the induced subgraph relation if and only if is an induced subgraph of . However, very little is known about well-quasi-ordered classes of graphs defined by more than one forbidden induced subgraph. We conjecture that for any natural number , there are finitely many minimal classes of graphs defined by forbidden induced subgraphs which are not well-quasi-ordered by the induced subgraph relation and prove the conjecture for . We explicitly reveal many of the minimal classes defined by two forbidden induced subgraphs which are not well-quasi-ordered and many of those which are well-quasi-ordered by the induced subgraph relation. [Copyright &y& Elsevier]

Abstract: Forming closures of subsets of a set X is a technique that plays an important role in many scientific disciplines and there are many cryptomorphic mathematical structures that describe closures and their construction. One of them was introduced by Aumann in the year 1970 under the name contact relation. Using relation algebra, we generalize Aumann’s notion of a contact relation between X and its powerset and that of a closure operation on from powersets to general membership relations and their induced partial orders. We also investigate the relationship between contacts and closures in this general setting and present some applications. In particular, we investigate the connections between contacts, closures and topologies and use contacts to establish a one-to-one correspondence between the column intersections space and the row intersections space of arbitrary relations. [Copyright &y& Elsevier]

Abstract: Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations. Ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for boolean functions. Among the many areas of application are verification, model checking, computer-aided design, relational algebra, and symbolic graph algorithms. In this paper it is shown that the OBDD complexity of the most significant bit of integer multiplication is exponential answering an open question posed by Wegener (2000) . [Copyright &y& Elsevier]

Abstract: Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainties. In this paper, first we prove that certain De Morgan’s law hold in soft set theory with respect to different operations on soft sets. Then, we discuss the basic properties of operations on soft sets such as intersection, extended intersection, restricted union and restricted difference. Moreover, we illustrate their interconnections between each other. Also we define the notion of restricted symmetric difference of soft sets and investigate its properties. The main purpose of this paper is to extend the theoretical aspect of operations on soft sets. [Copyright &y& Elsevier]

Abstract: The family of domain algebras provide an elegant formal system for automated reasoning about programme verification. Their primary models are algebras of relations, viz. representable domain algebras. We prove that, even for the minimal signature consisting of the domain and composition operations, the class of representable domain algebras is not finitely axiomatizable. Then we show similar results for extended similarity types of domain algebras. [ABSTRACT FROM AUTHOR]

Abstract: We introduce and study the notion of fuzzy multialgebras on the basis of a new definition of a fuzzy hyperoperation. We introduce the notion of fuzzy regular (resp., fuzzy strongly regular) relations of hyperalgebras and obtain their basic properties. Finally, we prove that with every fuzzy hyperalgebra we can associate a unique hyperalgebra via a regular (resp., strongly regular) relation. [ABSTRACT FROM AUTHOR]

Abstract: We consider orderings of nested subspaces of the space of Laurent polynomials on the real line, more general than the balanced orderings associated with the ordered bases and . We show that with such orderings the sequence of orthonormal Laurent polynomials determined by a positive linear functional satisfies a three-term recurrence relation. Reciprocally, we show that with such orderings a sequence of Laurent polynomials which satisfies a recurrence relation of this form is orthonormal with respect to a certain positive functional. [ABSTRACT FROM AUTHOR]

Abstract: Refinement is the notion of development between formal specifications. For specifications given in a relational formalism, downward and upward simulations are the standard method to verify that a refinement holds, their usefulness based upon their soundness and joint completeness. This is known to be true for total relational specifications and has been claimed to hold for partial relational specifications in both the non-blocking and blocking interpretations. In this paper we show that downward and upward simulations in the blocking interpretation, where domains are “guards”, are not jointly complete. This contradicts earlier claims in the literature. We illustrate this with an example (based on one recently constructed by Reeves and Streader) and then construct a proof to show why joint completeness fails in general. [ABSTRACT FROM AUTHOR]

Abstract: We elaborate our relational model of non-strict, imperative computations. The theory is extended to support infinite data structures. To facilitate their use in programs, we extend the programming language by concepts such as procedures, parameters, partial application, algebraic data types, pattern matching and list comprehensions. For each concept, we provide a relational semantics. Abstraction is further improved by programming patterns such as fold, unfold and divide-and-conquer. To support program reasoning, we prove laws such as fold–map fusion, otherwise known from functional programming languages. We give examples to show the use of our concepts in programs. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with a special timetabling problem. It was posed to us by the administration of our university and stems from the adoption of the British-American system of university education in Germany. This change led to the concrete task of constructing a timetable that enables the undergraduate education of secondary school teachers within three years in the “normal case” and within four years in the case of exceptional combinations of subjects. We develop two relation-algebraic models of the timetabling problem and in each case algorithms for computing solutions. The latter easily can be implemented in the Kiel RelView tool showing that RelView can be used for timetabling. [ABSTRACT FROM AUTHOR]

Abstract: Restating a unification problem as a single relational substitution instead of as multiple functional substitutions (or terms), a solution becomes a “determiniser” arrow and allows formalisation in the context of locally ordered categories with domain. This relies on the determinacy concept of “characterisation by domain” introduced by Desharnais and Möller for Kleene algebras with domain; this is here applied in the weakest possible setting. We show how “most general determinisers” can be seen as generalisation of quotient projections of partial equivalence relations, and show a characterisation that manages to avoid using converse or symmetry by employing restricted residuals instead. [ABSTRACT FROM AUTHOR]