Arrow allegories provide a convenient abstract framework to work with lattice-valued relations, or more precisely, relations that use the elements of a given Heyting algebra as truth values. One characteristic of arrow allegories is that all relations of the given arrow allegory use the same Heyting algebra ℋ. In this paper we want to extend this approach to allegories where relations between different objects may use different lattices of truth values and even further to relations that use a different lattice of truth values for every pair in the relation. Therefore, we define three concrete allegories, Rel(ℋ), Rel^u(ℋ) and ℋ-Rel, where the allegory listed later is a full suballegory of the previous ones. These three allegories capture the three different situations mentioned above. In particular, ℋ-Rel is the standard example of an arrow category. We investigate these allegories and provide suitable categorical definitions for these structures.

In this paper we are interested in the dual composition of two relations. The dual composition is defined to be the composition of relations on the dual order structure of a distributive allegory if it exists. We investigate this operation and the corresponding algebraic/categorical structures. In particular, we show that its existence guarantees that complemented relations are closed under composition, and we investigate when the dual composition of two univalent relations is again univalent.

Given a finite and non-empty set $X$ and randomly selected specific functions and relations on $X$, we investigate the existence and non-existence of fixed points and reflexive points, respectively. First, we consider the class of functions, weaken it to the classes of partial functions, total relations and general relations and also strengthen it to the class of permutations. Then we investigate the class of involutions and the subclass of proper involutions. Finally, we treat idempotent functions, partial idempotent functions and related concepts. We count relations, calculate corresponding probabilities and also calculate the limiting values of the latter in case that the cardinality of $X$ tends to infinity. All these results have been motivated and also supported by numerous experiments performed with the RelView tool.

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.

An equivalence relation can be constructed from a given (homogeneous, binary) relation in two steps: first, construct the smallest reflexive and transitive relation containing the given relation (the "star" of the relation) and, second, construct the largest symmetric relation that is included in the result of the first step. The fact that the final result is also reflexive and transitive (as well as symmetric), and thus an equivalence relation, is not immediately obvious, although straightforward to prove. Rather than prove that the defining properties of reflexivity and transitivity are satisfied, we establish reflexivity and transitivity constructively by exhibiting a starth root-in a way that emphasises the creative process in its construction. The resulting construction is fundamental to algorithms that determine the strongly connected components of a graph as well as the decomposition of a graph into its strongly connected components together with an acyclic graph connecting such components.

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.

Social choice theory investigates how individual preferences can be combined into a collective decision. An individual preference can be given by a preference relation indicating that the individual definitely prefers one alternative over the other, by a weak preference relation indicating that the individual prefers one alternative over the other or is indifferent between the alternatives, or by a choice relation indicating which alternative the individual would choose from a given set of alternatives. All three approaches are equivalent under certain assumptions on the corresponding relations. In this paper we investigate these relations in the context of L-fuzziness, i.e., the corresponding relations are fuzzy relations where the membership values are taken from a Heyting algebra L. Using the abstract theory of arrow categories we provide the most general setting in which these three approaches are equivalent.

We present a general method for computing vector representations 𝔯 of sets of relations. This method is used for obtaining 𝔯 from an inclusion ℜ⊆𝔖 , where ℜ and 𝔖 are relation-algebraic expressions over a relation rather than its vector representation. The core of the method is a theorem that shows how 𝔯 can be obtained from ℜ⊆𝔖 in one step. As applications we consider some problems concerning kernels of relations.

The purpose of this paper is to investigate topics in fuzzy concept analysis using the theory of arrow and fuzzy categories. Our approach deals with the original, fuzzy data using fuzzy subsets in order to obtain a notion of fuzzy concepts, fuzzy concept lattices, and attribute implications between fuzzy sets of attributes. Our approach is general in the sense that we only require a Heyting algebra together with a t-norm like operation as the lattice of truth values instead of the unit interval [0,1] or a substructure thereof. As a consequence, regular concept analysis, i.e., concept analysis based on regular sets and the Boolean truth values is just a special case of our approach. In addition, our approach differs from a lot of approaches in the literature, which usually first apply some kind of defuzzyfication, e.g., α-cuts, and then apply concept analysis in the classical (crisp) sense.

The purpose of this paper is to investigate topics in fuzzy concept analysis using the theory of arrow and fuzzy categories. Our approach deals with the original, fuzzy data using fuzzy subsets in order to obtain a notion of fuzzy concepts, fuzzy concept lattices, and attribute implications between fuzzy sets of attributes. Our approach is general in the sense that we only require a Heyting algebra together with a t-norm like operation as the lattice of truth values instead of the unit interval [ 0 , 1 ] or a substructure thereof. As a consequence, regular concept analysis, i.e., concept analysis based on regular sets and the Boolean truth values is just a special case of our approach. In addition, our approach differs from a lot of approaches in the literature, which usually first apply some kind of defuzzyfication, e.g., α -cuts, and then apply concept analysis in the classical (crisp) sense.

In this paper we start the investigation of an object representing the real numbers in categories of relations. Our axiomatization uses the construction of a relation power, i.e., an abstract version of power sets within the category. This allows us to utilize a relation algebraic version of Tarski’s axioms of the real numbers as a first-order definition of a real number object. The current paper focuses on the addition operation of the real number object. It is shown that addition forms a densely and linearly ordered abelian group.

Arrow categories establish a suitable framework to reason about L-fuzzy relation abstractly. For each arrow category we can identify the Heyting algebra L that is used as the lattice of membership or truth values by the relations of the category. Therefore, arrow categories model the fixed-base approach to L-fuzziness, i.e., all relations of the given arrow category use the same membership values. In this paper we are interested in the process of changing the base, i.e., an operation that allows to switch from an L-1-fuzzy relation to an L-2-fuzzy relation by replacing all membership values from L-1 by values from L-2. We will define and investigate this change of base between two abstract arrow categories for which component-wise reasoning cannot be performed.

Dedekind categories and similar structures provide a suitable framework to reason about binary relations in an abstract setting. Arrow categories extend this theory by certain operations and axioms so that additional aspects of L-fuzzy relations become expressible. In particular, arrow categories allow to identify crisp relations among all relations. On the other hand, the new operations and axioms in arrow categories force the category to be uniform, i.e., to be within a particular subclass of Dedekind categories. As an extension, arrow categories inherit constructions from Dedekind categories such as the definition of relational sums and splittings. However, these constructions are usually modified in arrow categories by requiring that certain relations are additionally crisp. This additional crispness requirement and the fact that the category is uniform raises a general question about these constructions in arrow categories. When can we guarantee the existence of the construction with and without the additional requirement of crispness in the given arrow category or an extension thereof? This paper provides a complete answer to this complex question for the two constructions mentioned.

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.

In this paper we focus on a fuzzy version of the so-called (un)sharpness property of relational products in arrow/fuzzy categories. It is shown that the fuzzy version can be reduced to a regular (un)sharpness problem. As a consequence we obtain that relational products are also sharp in the fuzzy sense if all relational products and powers exist. This result is important in applications of arrow/fuzzy categories since relational products, and, hence, the fuzzy version of the (un)sharpness problem, are integral components of these applications.

s of Invited Speakers Probabilistic Bisimulation with Silent Moves

s of Invited Speakers Probabilistic Bisimulation with Silent Moves

Using concepts from order theory and graph theory we investigate the dimensions $$ ind $$ , $$ Ind $$ and $$ dim $$ of finite topological spaces and Alexandroff spaces. We present specifications of them by means of specialisation pre-orders and algorithms for their computation. For finite spaces we give sharp upper bounds, characterisations of maximal-dimensional spaces via specialisation pre-orders and determine the number of maximal-dimensional spaces on a given set and whether these spaces are homeomorphic. These questions are also investigated for zero-dimensional Alexandroff spaces. We also consider relationships between the dimensions $$ ind $$ , $$ Ind $$ and $$ dim $$ .

System-of-systems engineering and related models of engineering design require intertwining of structures, respectively, at least for components in products, engineering and business activities, and people involved in those activities as also specifically connected with production and its management. In this paper we describe the logical machinery of system-of-systems engineering and related models of engineering design in sufficient so as to enable to describe the algebraic foundation of the many-valued logic that is inherent in these systems and structures. We are thus essentially unravelling the hidden and underlying logic of these systems and their related information and process structures, focusing on interaction between elements of the system.