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In this paper, considering L a complete residuated lattice, we present a lattice-valued category TNG (resp., TTNG) of (topological) ⊤-neighborhood groups, where the object is defined as a group equipped with a (topological) ⊤-neighborhood space such that the group operations are continuous with respect to the (topological) ⊤-neighborhood space. It is proved that: (1) The ⊤-neighborhood space associated with a ⊤-neighborhood group is topological, so the category TNG is equivalent to the category TTNG. Hence TTNG is redundant, and we need only discuss TNG. (2) The category TNG has nice characterizations, localization and uniformization. (3) The category TNG has initial structure, so it is a topological category, and each initial structure has an ordered representation. (4) The category NG of neighborhood groups can be embedded in TNG as a reflective subcategory. (5) The category TNG can be embedded in the category SLNG of stratified L -neighborhood groups as a reflective subcategory when the underlying lattice L is a meet-continuous lattice. (6) The category TNG is equivalent to the category StrLTOPG of strong L -topological groups when L is a complete MV-algebra. [ABSTRACT FROM AUTHOR]

Starting with an approximation space as the underlying structure, we look at the rough uniformity of a topological rough group. Next, taking L as a complete residuated lattice, we consider L -subgroup and normal L -subgroup of a group to create the L -fuzzy upper rough subgroup, and the L -fuzzy lower rough subgroup within the framework of the L -fuzzy approximation spaces. Here we particularly focus on a category of L -fuzzy upper rough subgroups, and a special kind of category of L -closure groups that arises naturally. We introduce the notion of the L -fuzzy approximation group, and study some of its properties including the usual function space structure for the L -fuzzy approximation spaces. Furthermore, using the notion of an L -fuzzy upper approximation operator, we investigate some categorical connection between the L -fuzzy approximation groups, and the L -closure groups. In a similar fashion, using an L -fuzzy lower approximation operator, we investigate the categorical connection between the L -fuzzy approximation groups, and the L -interior groups. [ABSTRACT FROM AUTHOR]

[Display omitted] Attention-based transformer neural networks have had significant impact in recent years. However, their applicability to model the behavior of physical systems has not yet been broadly explored. This is partly due to the high computational burden owing to the nonlinear scaling of very deep models, preventing application to a range of physical systems, in particular complex field data. Here we report the development of a general-purpose attention-based deep neural network model using a multi-headed self-attention approach, FieldPerceiver, that is capable of effectively predicting physical field data – such as stress, energy and displacement fields, as well as predicting overall material properties that characterize the statistics of stress distributions due to applied loading and crack defects, solely based on descriptive input that characterizes the material microstructure based on a set of interacting building blocks, all while capturing extreme short- and long-range relationships. Not using images as input, but rather realizing a neural olog description of materials where the categorization is learned by multi-headed attention, the model has no domain knowledge in its formulation, uses no convolutional layers, scales well to extremely large sizes and has no knowledge about specific properties of the material building blocks. Specifically, as applied to a fracture mechanics problem considered here, the model is capable of capturing size, orientation and geometry effects of crack problems for near- and far-field predictions, offering an alternative way to model materials failure based on language modeling without any convolutional layers commonly used in similar problems. We show that the FieldPerceiver can be used in a general framework, where the model can use insights learned during an initial, general training stage in order to fine-tune predictions for new scenarios, even when using only small additional datasets, revealing its broad generalization capacity. Once trained, the model can make predictions of thousands of scenarios within just a few minutes of compute time. It would take tens of hours, days or months to compute similar output using molecular dynamics simulation, for instance. [ABSTRACT FROM AUTHOR]

RobotCat uses a goal-oriented, declarative approach for robot programming that leverages mathematical representations found in category theory as a way to formally model modularity and behavioral knowledge. We define hierarchical interfaces that require only local modifications when new software or work cell models are introduced, thus making robot programming more interoperable. Note to Practitioners—RoboCat can be put into practice and be beneficial to different stakeholders: 1) a student (or small, individual manufacturer) is in charge of planning the production of a new product. Instead of tedious programming, she sketches out process diagrams using conceptual templates for actions (e.g., “move,” “carry,” “locate”) that can produce vastly more resilient production plans than would normally be expected from someone without many years of experience; 2) a robot platform manufacturer has a new manipulator that they would like to provide to the existing customers. Usually, such an upgrade requires a complex retooling of the fabrication lines to accommodate the new functionality. Further complicating matters, the new manipulator contains third-party hardware and software, requiring a full re-architecting of the integrated system. Rather than a wholesale swap of the old system for the new, the third-party application programming interfaces (APIs) can be systematically translated to fit snuggly within the existing modular system with seamless interoperability; and 3) a company is designing the production process associated with their new product launch and they wish to spread their supply chain across several manufacturing vendors. Rather than writing individual process plans customized for each manufacturer, they compose a high-level plan in a functional language. This plan is compiled into implementations specific to each manufacturer’s systems, without the need for human intervention. [ABSTRACT FROM AUTHOR]

This paper is a contribution to the search for efficient and high-level mathematical tools to specify and reason about (abstract) programming languages or calculi. Generalising the reduction monads of Ahrens et al., we introduce transition monads, thus covering new applications such as ...μ-calculus, λ-calculus, Positive GSOS specifications, differential λ-calculus, and the simply-typed, call-by-value λ-calculus. Moreover, we design a suitable notion of signature for transition monads. [ABSTRACT FROM AUTHOR]

The aim of the paper is to assess the relative merits of two formal representations of structure, namely, set theory and category theory. The purpose is to articulate ontic structural realism (OSR). In turn, this will facilitate a discussion on the strengths and weaknesses of both concepts, and will lead to a proposal for a pragmatics-based approach to the question of the choice of an appropriate framework. First, we present a case study from contemporary science—a comparison of the formulation of quantum mechanics in a language of Hilbert spaces and abstract C ⋆ -algebras. It is then shown how the method of structural representation can be determined based on the pragmatics of goal-oriented research, not a dogmatic choice. We investigate a hypothesis stating that use of the interplay between the powers of abstraction and detail of different representational methods results in adopting a pluralistic, as opposed to standard, unificatory, perspective on the role of structural representation in OSR. [ABSTRACT FROM AUTHOR]

The article proposes the use of geospatial information to support managerial decision-making by public authorities in the field of reintegration of temporarily occupied territories in Donetsk, Luhansk regions and Crimea. Variants of application of decision support systems in the management of immovable military property of the Armed Forces of Ukraine, the existing algorithms and methods of the decision support system in land relations are analyzed. Proposals are presented to support the adoption of managerial decisions by public authorities on the basis of Earth remote sensing data. The algorithm for the functioning of the decision support system in solving the problems of reintegration of temporarily occupied territories has been improved. The proposed sequence of actions of the method of the process of supporting the adoption of administrative decisions by public authorities in the field of reintegration of temporarily occupied territories using geospatial information, mathematical constructions of category theory and predicate logic. The order of implementation of the sequence of actions of this method is given. By improving and detailing the proposed method, it is possible to form a more effective system for supporting the adoption of managerial decisions by public authorities using geospatial information

Abstract The aim of this work is to argue for the idea that processes and process-based entities are to be modelled as relational structures. Relational structures are genuine structures, namely entities not committed to the existence of basic objects. My argument moves from the analysis of Whitehead’s original insight about process-based entities that, despite some residual of substance metaphysics, has the merit of grounding the intrinsic dynamism of reality on the holistic and relational characters of process-based entities. The current model of process ontology requires genuine emergence and this, in turn, requires organizations, i.e., emergence in organizations. Another view about processes rely on a structural specification of processes. I suggest that the two views can be made compatible by the help of a specific sort of structures, namely relational structures. The appeal to the mathematical theory of genuine structures, category theory, reveals the formal plausibility of this convergence. According to this formal approach, genuine structures are essentially dynamic entities for they are relational, namely, as well as organizations, they are not existentially committed to particulars.

The support for multi-model data has become a standard for most of the existing DBMSs. However, the step from a conceptual (e.g., ER or UML) schema to a logical multi-model schema of a particular DBMS is not straightforward. In this paper, we extend our previous proposal of multi-model data representation using category theory for transformations between models. We introduce a mapping between multi-model data and the categorical representation and algorithms for mutual transformations between them. We also show how the algorithms can be implemented using the idea of wrappers with the interface published but specific internal details concealed. Finally, we discuss the applicability of the approach to various data management tasks, such as conceptual querying. [ABSTRACT FROM AUTHOR]

On the one hand, the extension of ordinary institution theory, known as the theory of stratified institutions, is a general axiomatic approach to model theories where the satisfaction is parameterized by states of the models. On the other hand, the theory of 3 / 2 -institutions is an extension of ordinary institution theory that accommodates the partiality of the signature morphisms and its syntactic and semantic effects. The latter extension is motivated by applications to conceptual blending and software evolution. In this paper, we develop a general representation theorem of 3 / 2 -institutions as stratified institutions. This enables a transfer of conceptual infrastructure from stratified to 3 / 2 -institutions. We provide some examples in this direction. [ABSTRACT FROM AUTHOR]

Copyright of Proceedings of the Estonian Academy of Sciences is the property of Teaduste Akadeemia Kirjastus and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Annals of Philosophy / Roczniki Filozoficzne is the property of John Paul II Catholic University of Lublin, Faculty of Philosophy and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper develops a framework based on category theory which unifies the simultaneous consideration of timepoints, metrical relations, and meter inclusion founded on the category R e l of sets and binary relations. Metrical relations are defined as binary relations on the set of timepoints, and the subsequent use of the monoid they generate and of the corresponding functor to R e l allows us to define meter networks, i.e. networks of timepoints (or sets of timepoints) related by metrical relations. We compare this to existing theories of metrical conflict, such as those of Harald Krebs and Richard Cohn, and illustrate that these tools help to more effectively combine displacement and grouping dissonance and reflect analytical claims concerning nineteenth-century examples of complex hemiola and twentieth-century polymeter. We show that meter networks can be transformed into each other through meter network morphisms, which allows us to describe both meter displacements and meter inclusions. These networks are applied to various examples from the nineteenth and twentieth century. [ABSTRACT FROM AUTHOR]

Traditional transformational music theory describes transformations between musical elements as functions between sets and studies their subsequent algebraic properties and their use for music analysis. This is formalized from a categorical point of view by the use of functors C → S e t s where C is a category, often a group or a monoid. At the same time, binary relations have also been used in mathematical music theory to describe relations between musical elements, one of the most compelling examples being Douthett's and Steinbach's parsimonious relations on pitch-class sets. Such relations are often used in a geometrical setting, for example through the use of so-called parsimonious graphs to describe how musical elements relate to each other. This article examines a generalization of transformational approaches based on functors C → R e l , called relational presheaves, which focuses on the algebraic properties of binary relations defined over sets of musical elements. While binary relations include the particular case of functions, they provide additional flexibility as they also describe partial functions and allow the definition of multiple images for a given musical element. Our motivation to expand the toolbox of transformational music theory is illustrated in this paper by practical examples of monoids and categories generated by parsimonious and common-tone cross-type relations. At the same time, we describe the interplay between the algebraic properties of such objects and the geometrical properties of graph-based approaches. [ABSTRACT FROM AUTHOR]

A detailed study of graded frame, graded fuzzy topological system and fuzzy topological space with graded inclusion is already done in our earlier paper. The notions of graded fuzzy topological system and fuzzy topological space with graded inclusion were obtained via fuzzy geometric logic with graded consequence. As an offshoot, the notion of graded frame has been developed. This paper deals with a categorical study of graded frame, graded fuzzy topological system and fuzzy topological space with graded inclusion and their interrelation. [ABSTRACT FROM AUTHOR]

In mechatronic collaborative design, there is a synergic integration of several expert domains, where heterogeneous knowledge needs to be shared. To address this challenge, ontology-based approaches are proposed as a solution to overtake this heterogeneity. However, dynamic exchange between design teams is overlooked. Consequently, parametric-based approaches are developed to use constraints and parameters consistently during collaborative design. The most valuable knowledge that needs to be capitalized, which we call crucial knowledge, is identified with informal solutions. Thus, a formal identification and extraction is required. In this paper, we propose a new methodology to formalize the interconnection between stakeholders and facilitate the extraction and capitalization of crucial knowledge during the collaboration, based on the mathematical theory 'Category Theory' (CT). Firstly, we present an overview of most used methods for crucial knowledge identification in the context of collaborative design as well as a brief review of CT basic concepts. Secondly, we propose a methodology to formally extract crucial knowledge based on some fundamental concepts of category theory. Finally, a case study is considered to validate the proposed methodology. [ABSTRACT FROM AUTHOR]

Copyright of Japanese Journal of Cultural Anthropology / Bunka Jinruigaku is the property of Japanese Society of Cultural Anthropology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Book review: Category Theory in Physics, Mathematics, and Philosophy, Kuś M., Skowron B. (eds.), Springer Proc. Phys. 235, 2019, pp.xii+134.

“Is logic a physical variable?” This thought-provoking question was put forward by Michael Heller during the public lecture “Category Theory and Mathematical Structures of the Universe” delivered on 30th March 2017 at the National Quantum Information Center in Sopot. It touches upon the intimate relationship between the foundations of physics, mathematics and philosophy. To address this question one needs a conceptual framework, which is on the one hand rigorous and, on the other hand capacious enough to grasp the diversity of modern theoretical physics. Category theory is here a natural choice. It is not only an independent, well-developed and very advanced mathematical theory, but also a holistic, process-oriented way of thinking.

It is well-established that topos theory is inherently connected with intuitionistic logic. In recent times several works appeared concerning so-called complement-toposes (co-toposes), which are allegedly connected to the dual to intuitionistic logic. In this paper I present this new notion, some of the motivations for it, and some of its consequences. Then, I argue that, assuming equivalence of certain two definitions of a topos, the concept of a complement-classifier (and thus of a co-topos as well) is, at least in general and within the conceptual framework of category theory, not appropriately defined. For this purpose, I first analyze the standard notion of a subobject classifier, show its connection with the representability of the functor Sub via the Yoneda lemma, recall some other properties of the internal structure of a topos and, based on these, I critically comment on the notion of a complement-classifier (and thus of a co-topos as well).

Abstract In this article, we review ability of state-of-the-art ontology visualization tools for logical expressions. Then we propose an ontology visualization method with the goal of developing a complete and convenient visualization method, named logical graphs. The method is intended to represent the semantics of ontological structures, formulated as logical axioms of description, and it must use existing visualization methods from mathematical theories, such as Ch. S. Pierces Existential graphs, category theory, and graph theory. The proposed system is sufficient to describe the logic of ALC, SHIF, and SHOIN and visualize the OWL-Lite and OWL-DL ontologies to help casual users easily understand the ontology, as well as analyze data annotations and other tasks.

Competitiveness results from factors beyond the structural conditions and organizational boundaries, such as interorganizational cooperation. Evidence gathered in Brazilian credit unions suggests there is a social process in the firm for generating organizational capabilities and economic goods to satisfy needs defined by social structures in charge of solving market failures and structural deficiencies in the productive system. An exploratory and descriptive multiple case study with longitudinal qualitative data implemented using a systematic, computer-supported Process Tracing technique can refine a mathematical model of social phenomenon relying on a sequence of decision-making events of deterministic nature. Two case studies revealed that firms maintain relationships of cooperation and contribution with partners and structures in their organizational and environmental surroundings whenever their economic performance is constrained by competitive problems of systemic nature, for which there is no solution based on the mechanisms of market price and state intervention.

Using the design method proposed by Kunii, this study mathematically explains the achievement of Kublai of merging two different societies to build a globalized nation after conquering China. The considered design method, called incrementally modular abstraction hierarchy, consists of seven levels, from the most abstract, homotopy, to the most concrete, computer graphics. Moreover, it allows a flexible design while moving up and down its hierarchy. Using this method, we attempt to restore the concepts of Kublai, aiming to build a globalized state by combining nomadic and agricultural cultures. Specifically, we examine the use of two capitals to resolve the problem of different lifestyles of the Mongolian Khan, who moved between the summer and winter camps of the nomads, and the Chinese Emperor, who lived in a fixed palace of the agricultural people. By incorporating the category theory into Kunii's method, we discuss the adjunction of the Mongolian 1000-household social and military system based on decimal numbers with the Chinese bureaucratic system. The inclusion of the category theory in his original method allows visualization by programming, making the design more versatile and flexible. [ABSTRACT FROM AUTHOR]

In this paper, we studied the category of EQ-algebras and showed that it is complete, but it is not cocomplete, in general. We proved that multiplicatively relative EQ-algebras have coequlizers and we calculated coproduct and pushout in a special case. Also, we constructed a free EQ-algebra on a singleton. [ABSTRACT FROM AUTHOR]

Following the earlier formalism of the categorical representation learning, we discuss the construction of the ‘RG-flow-based categorifier’. Borrowing ideas from the theory of renormalization group (RG) flows in quantum field theory, holographic duality, and hyperbolic geometry and combining them with neural ordinary differential equation techniques, we construct a new algorithmic natural language processing architecture, called the RG-flow categorifier or for short the RG categorifier, which is capable of data classification and generation in all layers. We apply our algorithmic platform to biomedical data sets and show its performance in the field of sequence-to-function mapping. In particular, we apply the RG categorifier to particular genomic sequences of flu viruses and show how our technology is capable of extracting the information from given genomic sequences, finding their hidden symmetries and dominant features, classifying them, and using the trained data to make a stochastic prediction of new plausible generated sequences associated with a new set of viruses which could avoid the human immune system.

Explainable artificial intelligence (XAI) is a growing field that aims to increase the transparency and interpretability of machine learning (ML) models. The aim of this work is to use the categorical properties of learning algorithms in conjunction with the categorical perspective of the information in the datasets to give a framework for explainability. In order to achieve this, we are going to define the enriched categories, with decorated morphisms, $\pmb{\mathcal{Learn}}$ , $\pmb{\mathcal{Para}}$ and $\pmb{\mathcal{MNet}}$ of learners, parameterized functions, and neural networks over metric spaces respectively. The main idea is to encode information from the dataset via categorical methods, see how it propagates, and lastly, interpret the results thanks again to categorical (metric) information. This means that we can attach numerical (computable) information via enrichment to the structural information of the category. With this, we can translate theoretical information into parameters that are easily understandable. We will make use of different categories of enrichment to keep track of different kinds of information. That is, to see how differences in attributes of the data are modified by the algorithm to result in differences in the output to achieve better separation. In that way, the categorical framework gives us an algorithm to interpret what the learning algorithm is doing. Furthermore, since it is designed with generality in mind, it should be applicable in various different contexts. There are three main properties of category theory that help with the interpretability of ML models: formality, the existence of universal properties, and compositionality. The last property offers a way to combine smaller, simpler models that are easily understood to build larger ones. This is achieved by formally representing the structure of ML algorithms and information contained in the model. Finally, universal properties are a cornerstone of category theory. They help us characterize an object, not by its attributes, but by how it interacts with other objects. Thus, we can formally characterize an algorithm by how it interacts with the data. The main advantage of the framework is that it can unify under the same language different techniques used in XAI. Thus, using the same language and concepts we can describe a myriad of techniques and properties of ML algorithms, streamlining their explanation and making them easier to generalize and extrapolate.