In a series of papers Klement et al. investigated discrete integrals such as the Choquet and Sugeno integral and their axiomatization. As part of their study they showed that universal integrals are based on semicopulas, and they provided lower and upper bounds of the integral operations based on a given semicopula. These real-valued resp. unit interval valued integrals can be considered as proper aggregation tool in the context of fuzzy sets. The aim of the current paper is to generalize this approach to so-called L-fuzzy sets and relations, i.e., fuzzy sets and relations that use an arbitrary Heyting algebra L as membership degree instead of the unit interval. Furthermore, we present the theory within arrow categories, i.e., we abstract from concrete sets and relations and work within a suitable algebraic framework. The current paper also shows that the results of the previous work can be proven without referring to the real numbers and specific measures such as the Lebesque measure and the induced measurable spaces.

In this paper we continue the investigation of a real number object, i.e., an object representing the real numbers, in categories of relations. Our axiomatization is based on a relation algebraic version of Tarski's axioms of the real numbers. It was already shown that the addition of such an object forms a dense, linear ordered abelian group. In the current paper we will focus on the least-upper-bound property of such an object.

Finite topological spaces and their dimensions have many applications in computer science, e.g., in digital topology, computer graphics and the analysis and synthesis of digital images. Georgiou et. al. [11] provided a polynomial algorithm for computing the covering dimension dim( X; 𝒯) of a finite topological space (X; 𝒯). In addition, they asked whether algorithms of the same complexity for computing the small inductive dimension ind( X; 𝒯) and the large inductive dimension Ind( X; 𝒯) can be developed. The first problem was solved in a previous paper [4]. Using results of the that paper, we also solve the second problem in this paper. We present a polynomial algorithm for Ind( X; 𝒯), so that there are now efficient algorithms for the three most important notions of a dimension in topology. Our solution reduces the computation of Ind( X; 𝒯), where the specialisation pre-order of ( X; 𝒯) is taken as input, to the computation of the maximal height of a specific class of directed binary trees within the partially ordered set. For the latter an efficient algorithm is presented that is based on order- and graph-theoretic ideas. Also refinements and variants of the algorithm are discussed.

Despite significant effort, there is currently no formal or de facto standard framework or format for constructing, representing, or manipulating general neural networks. In computational neuroscience, there have been some attempts to formalize connectionist notations and generative operations for neural networks, including Connection Set Algebra, but none are truly formal or general. In computational intelligence (CI), though the use of linear algebra and tensor-based models are widespread, graph-based frameworks are also popular and there is a lack of tools supporting the transfer of information between systems. To address these gaps, we exploited existing results about the connection between linear and relation algebras to define a concise, formal algebraic framework that generalizes graph and tensor-based neural networks. For simplicity and compatibility, this framework is purposefully defined as a minimal extension to linear algebra. We demonstrate the merits of this approach first by defining new operations for network composition along with proofs of their most important properties. An implementation of the algebraic framework is presented and applied to create an instance of an artificial neural network that is compatible with both graph and tensor based CI frameworks. The result is an algebraic framework for neural networks that generalizes the formats used in at least two systems, together with an example implementation.

We introduce the decomposition of an arbitrary relation into a sequential composition of three relations, viz. of a mapping with a partial order and then the transpose of a mapping. After presenting some basic properties, we investigate the specific classes of junkfree, irreducible and minimal decompositions and show that for all relations a minimal decomposition exists. We also study decompositions with regard to DedekindMacNeille completions and concept lattices. These constructions are closely related to decompositions of relations. In our setting the fundamental theorem of concept lattices states that concept lattices are minimal-complete decompositions and all such decompositions are isomorphic. As a further main result we prove that the cutDedekindMacNeille completion of the order that belongs to the minimal decomposition of a relation is isomorphic to the concept lattice of that relation. Instead of considering binary relations on sets, we will work point-free within the general framework of allegories. This complement-free approach implies that the results of the paper can be applied to all models of these algebraic structures, including, for instance, lattice-valued fuzzy relations.

Timed contact algebras constitute an approach to a temporal version of a region based theory of space. The general theory does not provide a notion of an underlying static world, i.e. it does not explicitly contain a set of non moving regions. Furthermore, the model of time does not have any structure, i.e. time is neither ordered nor required to be discrete or continuous. In this paper we want to investigate two extensions of the basic theory. The first extension considers grounded timed contact algebras that make the underlying static world explicit. In this context we introduce the Axiom of Construction that relates the existence of certain regions and the time structure for the first time. The second addition is given by a betweenness relation on the set of time. In this context we introduce the Axiom of Continuity (CONT), ensuring "smooth'' movement of regions through time. Last but not least, we show that both axioms together do not allow finite models.

Boolean contact algebras constitute a convenient approach to a region based theory of space. In this paper we want to extend this approach to regions moving in time - called timed contact structures. We study their canonical models using topological spaces. As the main contribution we prove a general representation theorem for this kind of algebras.