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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: 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: Basic concepts of classical dynamics are analysed in the simple mathematical setting of state transition systems, where both time and space are discrete, and no structure is assumed on the state space besides a binary transition relation. This framework proves useful to the dynamical analysis of computations and biomolecular processes. Here a relational formulation of this framework is presented, where the concepts of attractor and recurrence surface in two variants, respectively relating to the two fundamental modalities. A strong link between recurrence and both existence and extent of attractors, in either variant, is established by a novel characterization theorem. Further concepts are easily casted in the relational language, such as product dynamics and projections thereof, which support analysis and reasoning about metabolic P systems. An outline of possible applications and future developments of this work concludes the article. [Copyright &y& Elsevier]

Abstract: The relational product construction is often consider as an abstract version of cartesian products. The existence of those products is strongly connected with the representability of that category. In this paper we investigate a canonical weakening of the notion of a relational product. Unlike the strong version, any (small) category of relations can be embedded into a suitable category providing all weak relational products. Furthermore, we provide several examples, and we study the categorical properties of the new construction. [Copyright &y& Elsevier]

Abstract: The importance of relational methods in temporal and spatial reasoning has been widely recognised in the last two decades. A quite large part of contemporary spatial reasoning is concerned with the research of relation algebras generated by the “part of” and “connection” relations in various domains. This paper is devoted to the study of one particular relation algebra appeared in the literature, viz. the complemented disk algebra. This algebra was first described by Düntsch [I. Düntsch, A tutorial on relation algebras and their application in spatial reasoning, Given at COSIT, August 1999, Available from: ] and then, Li et al. [Y. Li, S. Li, M. Ying, Relational reasoning in the Region Connection Calculus, Preprint, 2003, Available from: http://arxiv.org/abs/cs/0505041] showed that closed disks and their complements provides a representation. This set of regions is rather restrictive and, thus, of limited practical values. This paper will provide a general method for generating representations of this algebra in the framework of Region Connection Calculus. In particular, connected regions bounded by Jordan curves and their complements is also such a representation. [Copyright &y& Elsevier]

Abstract: Given a finite set E, a relation with base is a mapping such that for all . The restriction of R to a subset X of E is R considered as a mapping from into . Given an integer and a relation R with base E, we call -reconstruction of R every relation with the same base, any restriction of which to a subset X of E with up to k elements is isomorphic to the restriction of R to X. The relation R is -reconstructible when each -reconstruction of R is isomorphic to R. In this work, the structure of the non--reconstructible relations is studied for all . This study leads to an introduction of the minimal non--reconstructible relations and to characterization of the class of such relations for all . This class includes in particular the class of the indecomposable non--reconstructible relations. [Copyright &y& Elsevier]

Relation algebras were invented by Tarski and his collaborators in the middle of the 20th century. The concept of integrality arose naturally early in the history of the subject, as did various constructions of finite integral relation algebras. Later the concept of finite-dimensionality was introduced for classifying nonrepresentable relation algebras. This concept is closely connected to the number of variables used in proofs in first-order logic. Some results on these topics are presented in chronological order. [Copyright &y& Elsevier]