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This work focuses on establishing the relationship between bipolar fuzzy automata, reverse bipolar fuzzy automata, and double bipolar fuzzy automata. It also introduces the concepts of bipolar fuzzy subsystems, reverse bipolar fuzzy subsystems, and double bipolar fuzzy subsystems, and explores various properties associated with these subsystems. Furthermore, the paper aims to introduce the categorical aspects of bipolar fuzzy automata and reverse bipolar fuzzy automata, along with their functorial relationship. [ABSTRACT FROM AUTHOR]

This paper addresses L-fuzzy topologies induced by L-G-filters and studies the categorical relations between L-G-filter spaces and L-fuzzy topological spaces. Three functors from the category of L-G-filter spaces to the category of Lfuzzy topological spaces are obtained. Having introduced the concept of monotone L-fuzzy topologies, the study inquires into the sum, subspace, product, quotient and the lattice structure of such topologies. [ABSTRACT FROM AUTHOR]

Let p ∈ [1, ∞] and F : Set → Set be a functor with finite supports in the category Set of sets. Given a non-empty metric space (X, dX), we introduce the distance on the functor-space FX as the largest distance such that for every n ∈ ℕ and a ∈ Fn the map Xn → FX, f → Ff(a), is non-expanding with respect to the ℓp-metric on Xn. We prove that the distance is a pseudometric if and only if the functor F preserves singletons; is a metric if F preserves singletons and one of the following conditions holds: (1) the metric space (X, dX) is Lipschitz disconnected, (2) p = 1, (3) the functor F has finite degree, (4) F preserves supports. We prove that for any Lipschitz map f : (X, dX) → (Y, dY) between metric spaces the map is Lipschitz with Lipschitz constant Lip(Ff) ≤ Lip(f). If the functor F is finitary, has finite degree (and preserves supports), then F preserves uniformly continuous function, coarse functions, coarse equivalences, asymptotically Lipschitz functions, quasi-isometries (and continuous functions). For many dimension functions we prove the formula dim FpX ≤ deg(F) dim X. Using injective envelopes, we introduce a modification of the distance and prove that the functor Dist → Dist, , in the category Dist of distance spaces preserves Lipschitz maps and isometries between metric spaces. [ABSTRACT FROM AUTHOR]

We prove a classification of additive polynomial superfunctors, which allows us to compute some extensions of a superfunctor of the form F ∘ A where F is a classical polynomial functor and A is additive. We get a formula which relates these extensions to the classical ones of F. A possible generalisation is conjectured at the end. [ABSTRACT FROM AUTHOR]

In this paper, we have established the notions of soft order relations and studied its basic structural properties. We give the concepts of soft maximum, soft minimum, soft maximal, soft minimal, soft infimum and soft supremum in any s-poset. Then, the concept of soft order preserving mapping is defined and given some basic results. Moreover, it has been shown that the topology derived from a soft partial order relation is a soft topology which will substitute as a soft version of Alexandroff topology in classical theory. With all that, we obtained the category SPOSET of s-posets, and constructed a functor which is called ostracizer functor between the categories SPOSET and POSET . [ABSTRACT FROM AUTHOR]

We introduce certain functors from the category of commutative rings (and related categories) to that of Z -algebras (not necessarily associative or commutative). One of the motivating examples is the Leavitt path algebra functor R ↦ L R (E) for a given graph E. Our goal is to find "descending" isomorphism results of the type: if F , G are algebra functors and K ⊂ K ′ a field extension, under what conditions an isomorphism F (K ′) ≅ G (K ′) of K ′ -algebras implies the existence of an isomorphism F (K) ≅ G (K) of K-algebras? We find some positive answers to that problem for the so-called "extension invariant functors" which include the functors associated with Leavitt path algebras, Steinberg algebras, path algebras, group algebras, evolution algebras and others. For our purposes, we employ an extension of the Hilbert's Nullstellensatz Theorem for polynomials in possibly infinitely many variables, as one of our main tools. We also remark that for extension invariant functors F , G , an isomorphism F (H) ≅ G (H) , for some K-algebra H endowed with an augmentation, implies the existence of an isomorphism F (S) ≅ G (S) for any commutative and unital K-algebra S. [ABSTRACT FROM AUTHOR]

Notions such as śūnyata, catuskoti, and Indra's net, which figure prominently in Buddhist philosophy, are difficult to readily accommodate within our ordinary thinking about everyday objects. Famous Buddhist scholar Nāgārjuna considered two levels of reality: one called conventional reality, and the other ultimate reality. Within this framework, śūnyata refers to the claim that at the ultimate level objects are devoid of essence or 'intrinsic properties', but are interdependent by virtue of their relations to other objects. Catuskoti refers to the claim that four truth values, along with contradiction, are admissible in reasoning. Indra's net refers to the claim that every part of a whole is reflective of the whole. Here we present category theoretic constructions that are reminiscent of these Buddhist concepts. The universal mapping property definition of mathematical objects, wherein objects of a universe of discourse are defined not in terms of their content, but in terms of their relations to all objects of the universe is reminiscent of śūnyata. The objective logic of perception, with perception modeled as [a category of] two sequential processes (sensation followed by interpretation), and with its truth value object of four truth values, is reminiscent of the Buddhist logic of catuskoti. The category of categories, wherein every category has a subcategory of sets with zero structure within which every category can be modeled, is reminiscent of Indra's net. Our thorough elaboration of the parallels between Buddhist philosophy and category theory can facilitate better understanding of Buddhist philosophy, and bring out the broader philosophical import of category theory beyond mathematics. [ABSTRACT FROM AUTHOR]

The note deals with categories whose objects are functions from sets to -algebras and morphisms are -homomorphisms of -algebras making appropriate diagrams commutative. In the theory of universal -algebras, such categories satisfying certain additional axioms are called -relations. Those -relations that determine universal -algebras are said to be compact. In this note, we construct functors between compact -relations. These functors arise from -homomorphisms between universal -algebras which are determined by compact -relations. In the case when a functor is defined by an isomorphism of the universal -algebras, we show that this functor is an isomorphism of compact -relations. Moreover, we consider -relations which are called -polynomial relations associated with -polynomial pairs. It is shown that every -algebra is the universal -algebra generated by a -polynomial pair. As a consequence, we obtain that every compact -relation is isomorphic to a -polynomial relation. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the action for topological groupoids. In our work, some definitions and propositions are introduced and in addition to the basic construction of topological groupoid which are needed and we studied the different properties of categories and the properties of topological groupoid, and we define concepts such as topos, functor and sheave. [ABSTRACT FROM AUTHOR]

Learning to learn is a reduction in the amount of training neededfor task attainment across a series of similar tasks. Transferdifferentiates (adult) humans from other species, portending awindow into unique aspects of human learning. However, itsunclear whether such differences are quantitative, or qualitativeand what it means for the evolution/development of cognition.In this paper, learning is regarded as a (categorical) limit. Alimit is a universal construction, and so transfer follows froma (generalized) optimization process. This result provides aformal basis for comparison/contrast of learning transfer in hu-mans and other species—another step to bringing the empiricalquestion into sharper relief.

This paper presents the idea of Alexandrov L-quasi-G-filter space and examines its relationship with L-fuzzy relations. It is proved that every Alexandrov L-quasi-G-filter induces an L-fuzzy relation and vice-versa. By identifying certain functors, the study has brought out the connections between the categories of Alexandrov L-quasi-G-filter spaces and Alexandrov L-fuzzy pre-uniform spaces. Further, the study has explored and thereby establishes the scope of Alexandrov L-quasi-G-filter spaces in mathematical modeling and decision-making processes. [ABSTRACT FROM AUTHOR]

We are dealing with projective classes (in short |$\textrm {PC}$|⁠) over first-order vocabularies with no restrictions on the (possibly infinite) arities of relation or operation symbols. We verify that |$\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })=\textrm {RPC}(\mathbin {\mathscr {L}}_{\infty \lambda })$| for any infinite cardinal |$\lambda $|⁠ , and that if |$\lambda $| is singular, then |$\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })=\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda ^+})$|⁠. If |$\lambda $| is regular, then a class of structures over a |$\lambda $| -ary vocabulary is |$\textrm {PC}(\mathbin {\mathscr {L}}_{\infty \lambda })$| -definable iff it is the image of a |$\lambda $| -continuous functor on a |$\lambda $| -accessible category; we also provide separating counterexamples for the non |$\lambda $| -ary case. We prove that many |$\textrm {PC}$| classes of structures, previously known not to be closed under elementary equivalence over any |$\mathbin {\mathscr {L}}_{\infty \lambda }$|⁠ , are not even |$\textrm {co}\textrm {-}\textrm {PC}$| over |$\mathbin {\mathscr {L}}_{\infty \infty }$|⁠. Those classes arise from diverse contexts including convex |$\ell $| -subgroup lattices of lattice-ordered groups, ideal lattices of rings, nonstable K |$_0$| -theory of rings, coordinatization of sectionally complemented modular lattices and real spectra of commutative unital rings. For example, the class of posets of finitely generated two-sided ideals of all unital rings is |$\textrm {PC}$| but not |$\textrm {co}\textrm {-}\textrm {PC}$| over |$\mathbin {\mathscr {L}}_{\infty \infty }$|⁠. We also provide a negative solution to a problem, raised in 2011 by Gillibert and the author, asking whether essential surjectivity of a 'well-behaved' functor on objects entails its essential surjectivity on diagrams indexed by arbitrary finite posets. [ABSTRACT FROM AUTHOR]

Algebraic models of connected homotopy 3‐types such as quadratic modules, 2‐crossed modules, crossed squares, and their relations are studied in various ways. In this work, we obtain another natural equivalence for quadratic modules. That is, we define functors between quadratic modules and our candidate category. [ABSTRACT FROM AUTHOR]

Despite a general recognition that there are two styles of think-ing: fast, reflexive and relatively effortless (Type 1) versus slow,reflective and effortful (Type 2), dual-process theories of cogni-tion remain controversial, in particular, for their vagueness. Toaddress this lack of formal precision, we take a mathematicalcategory theory approach towards understanding what under-lies the relationship between dual cognitive processes. Fromour category theory perspective, we show that distinguishingfeatures of Type 1 versus Type 2 processes are exhibited viaadjoint and representable functors. These results suggest thatcategory theory provides a useful formal framework for devel-oping dual-process theories of cognition.

Category theorists and cognitive scientists study the structural (analogical) relations between domains of interest albeit in different contexts, that is, formal and psychological systems, respectively. Despite this basic commonality, very few cognitive scientists take a category theory approach toward understanding the structure of cognition which raises the question, What is category theory to cognitive science? An answer is given as the slogan "Category theory is to cognitive science as functor is to representation; as natural transformation is to comparison" to make category theory more accessible and informative for cognitive scientists. [ABSTRACT FROM AUTHOR]

Although traditional and idempotent mathematics are "parallel'', by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be "parallel''. At first we establish for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic ("parallelism''). Then we construct an example which shows that the constructions P and I form distinguished functors from each other ("parallelism'' negation). Further for a compact Hausdorff space X we establish that the hereditary normality of I3(X)\ X implies the metrizability of X.

Category theorists and cognitive scientists study the structural (analogical) relations between domains of interest albeit in different contexts, that is, formal and psychological systems, respectively. Despite this basic commonality, very few cognitive scientists take a category theory approach toward understanding the structure of cognition which raises the question, What is category theory to cognitive science? An answer is given as the slogan “Category theory is to cognitive science as functor is to representation; as natural transformation is to comparison” to make category theory more accessible and informative for cognitive scientists.

In this paper, the authors introduce catalyzed LM-G-filter spaces, a special case of weakly inspired LM-G-filter spaces and identify certain properties of these spaces. It is proved that CLM-G, the category of catalyzed LM-G-filter spaces, is isomorphic to ILM-G, the category of inspired LM-G-filter spaces. Moreover, the categorical connection between WILM-G, the category of weakly inspired LM-G-filter spaces, and CLM-G is investigated through interior and exterior catalyzation of weakly inspired LM-G-filter spaces. It is proved that CLM-G is an isomorphism-closed, bireflective and bicoreflective full subcategory of WILM-G and LM-G. [ABSTRACT FROM AUTHOR]

In the current study, by a general fuzzy automaton we aim at showing a set of propositions related to a given automaton showing that the truth-values are depended on thestates, inputs and membership values of active states at time t. This new approach enables us to consider automata from a different point of view which is more close to logical treatment and helps us make estimations about the behavior of automaton particularly in a nondeterministic mode. The logic consists of propositions on the given GFA and its dynamic nature is stated by means of the so-called transition functor. This logic enables us to derive a certain relation on the set of states labeled by inputs. In fact, it is shown that if our set of propositions is large enough, this recovering of the transition relation is possible. Through a synthesis in the theory of systems, this study contributes to construct a general fuzzy automaton which realizes a dynamic process at least partially known to the user, which has been fully achieved in Theorem 3.6. Also, we study the theory of general fuzzy automata by using the concepts of operators. Such operators help us in the algebraic study of general fuzzy automata theory and provide a platform to use fuzzy topological therein. Further, a Galois connection is obtained between the state-transition relation on states and thetransition operators on propositions. To illustrate the proposed approach, the subject matter is more elaborated in detail through examples.

The cocompleteness of the category is proved. [ABSTRACT FROM AUTHOR]

Anti-elementarity is a strong way of ensuring that a class of structures, in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form ℒ ∞ λ . We prove that many naturally defined classes are anti-elementary, including the following: the class of all lattices of finitely generated convex ℓ -subgroups of members of any class of ℓ -groups containing all Archimedean ℓ -groups; the class of all semilattices of finitely generated ℓ -ideals of members of any nontrivial quasivariety of ℓ -groups; the class of all Stone duals of spectra of MV-algebras — this yields a negative solution to the MV-spectrum Problem; the class of all semilattices of finitely generated two-sided ideals of rings; the class of all semilattices of finitely generated submodules of modules; the class of all monoids encoding the nonstable K0-theory of von Neumann regular rings, respectively, C*-algebras of real rank zero; (assuming arbitrarily large Erdős cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large 4 -frame. The main underlying principle is that under quite general conditions, for a functor Φ : 𝒜 → ℬ , if there exists a noncommutative diagram D → of 𝒜 , indexed by a common sort of poset called an almost join-semilattice, such that Φ D → I is a commutative diagram for every set I , Φ D → ≇ Φ X → for any commutative diagram X → in 𝒜 , then the range of Φ is anti-elementary. [ABSTRACT FROM AUTHOR]

Although traditional and idempotent mathematics are "parallel", by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be "parallel". At first we establish for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic ("parallelism"). Then we construct an example which shows that the constructions P and I form distinguished functors from each other ("parallelism" negation). Further for a com-pact Hausdorff space X we establish that the hereditary normality of I3(X)\X implies the metrizability of X. [ABSTRACT FROM AUTHOR]

The objective of this paper is to establish the relationship between L-valued approximation spaces and L-valued transformation systems. We show that for each L-valued upper/lower fuzzy transformation system there exist an L-valued reflexive approximation space and vice versa. In between, we study the concept of L-valued natural transformations.

Of special relevance for the NP vs. DP debate are nominals in which the determiner has other values for number, gender, case and/or person than the noun, for depending on whether such nominals share these values with the noun or with the determiner they provide evidence in favor of either the NP or the DP approach. This paper discusses some nominals of this kind and shows that they share the relevant feature values with the noun. It then develops an NP analysis for these nominals which is cast in the notation of Head-driven Phrase Structure Grammar. It also shows what a DP analysis would look like. A comparison reveals that the NP analyses are less complex, more uniform and less stipulative. Examples are taken from Dutch, English, German, French, Italian, Spanish, Polish and Serbo-Croatian.

The paper considers open computing systems, which provide the necessary growth of performance and memory by mechanical addition of new units without affecting the existing software environment. Such computing systems are based on the application of a special functionally complete element base (planner, functor, communicator) that implements parallel processing using dataflow control flows when necessary program fragments are transferred along with the data. Aimed at this, when a certain procedure is found ready for starting (the planner is ready for all the data it needs), the corresponding part of the program is opened in the planner – the operator, which is then transferred along with the data to the free execution device – the functor. The result is always returned along the same route by which the procedure was activated. The layouts of the open systems using two units of design are considered: cells on the reduced element base, and servers assembled from these cells. A two-level distributed switching environment is used. At the cell level, it is provided by the transit properties of the planners and functors, and at the server level – by communicators that are part of the cells. Three types of cells are identified, which enable the growth of computing system functions: cells for increasing the number of gateways used to exchange with the external environment, cells for increasing control and working memory, cells for increasing performance. The recession of the concentrated switching environment allowed for these types of computing systems extensions to be performed independently and without any restrictions on their size. A three-dimensional structure of open systems is described, which can be used to build supercomputers.

This paper studies * -topology and s-topology in polysaturated nonstandard model,which are induced by metric space on two nonstandard sets. In order to construct * -topology , the set of finite points is introduced. It shows that every internal set is compact, every open set is saturated and standard part mapping is continuous. Finally,the closure operator,interior operator and mapping * are discussed under s-topology .

We construct a 2-categorical extension of the relative entropy functor of Baez and Fritz, and show that our construction is functorial with respect to vertical morphisms. Moreover, we show such a ‘2-relative entropy’ satisfies natural 2-categorial analogues of convex linearity, vanishing under optimal hypotheses, and lower semicontinuity. While relative entropy is a relative measure of information between probability distributions, we view our construction as a relative measure of information between channels.

The C++ Standard Template Library (STL) is the most well-known and widely used library that is based on the generic programming paradigm. STL takes advantage of C++ templates, so it is an extensible, effective and flexible system. Professional C++ programs cannot miss the usage of the STL because it increases quality, maintainability, understandability and efficacy of the code. However, the usage of C++ STL does not guarantee perfect, error-free code. Contrarily, incorrect application of the library may introduce new types of problems. Unfortunately, there is still a large number of properties that are tested neither at compilation-time nor at run-time. It is not surprising that in implementations of C++ programs so many STL-related bugs may occur. It is clearly seen that the compilation validation is not enough to exclude STL-related bugs. For instance, the mathematical properties of user-defined sorting parameters are not validated at compilation phase nor at run-time. Contravention of the strict weak ordering property results in weird behavior that is hard to debug. In this paper, we argue for a static analysis tool which finds erroneous implementation of functors regarding the mathematical properties. The primary goal is to support Continuous Integration pipelines, using this tool during development to overcome debugging efforts. [ABSTRACT FROM AUTHOR]

In our paper [3] we built the functor dExt n S-1A(-; S-1B) in the category A-Alg. The purpuse of this paper is to show that if A is a ring not necessary commutative, S a central multiplicatively closed subset of A and B an (A-A)-bialgebra, then TorS-1A n (-, S-1B) : Alg-S-1A = S-1A-Modo: Ext n S-1A(-, S-1B)o is an adjunction. [ABSTRACT FROM AUTHOR]

The objective of this paper is to establish the relationship between fuzzy approximation operators and fuzzy transformation systems. We show that for each upper fuzzy transformation system there exists a fuzzy reflexive approximation space and vice-versa. We further establish such relationship between lower fuzzy transformation systems and fuzzy reflexive approximation spaces under the condition that the underline lattice structure satisfies double negation law. [ABSTRACT FROM AUTHOR]

We survey the research on the inductive systems of -algebras over arbitrary partially ordered sets. The motivation for our work comes from the theory of reduced semigroup -algebras and local quantum field theory. We study the inductive limits for the inductive systems of Toeplitz algebras over directed sets. The connecting -homomorphisms of such systems are defined by sets of natural numbers satisfying some coherent property. These inductive limits coincide up to isomorphisms with the reduced semigroup -algebras for the semigroups of non-negative rational numbers. By Zorn's lemma, every partially ordered set is the union of the family of its maximal directed subsets indexed by elements of a set . For a given inductive system of -algebras over one can construct the inductive subsystems over and the inductive limits for these subsystems. We consider a topology on the set . It is shown that characteristics of this topology are closely related to properties of the limits for the inductive subsystems. [ABSTRACT FROM AUTHOR]

Let (R, RUS, SVR, S) be a Morita context. We study the connection between the classes of preradicals PR(R) and PR(S) of the categories of modules R-Mod and S-Mod. Two mappings between PR(R) and PR(S) are constructed from the functors HomR(U, -) and HomS(V, -). We investigate several properties of these mappings, in particular their behavior relative to the order, intersection and heredity. Similar methods were used earlier for preradicals in an adjoint situation. [ABSTRACT FROM AUTHOR]

In this paper, we examine categorical and cardinal properties of hyperspaces with finite number of components. We prove that the functor Cn : Comp → Comp is not normal, i.e., it does not preserve epimorphisms of continuous mappings. We also discuss the density, the caliber, and the Shanin number of the space Cn(X). [ABSTRACT FROM AUTHOR]

In this paper, we examine topological and dimensional properties of metric, Tychonoff, and compact C-spaces under the action of the covariant subfunctor Pf of the functor P of probability measures in the category of metric, compact, and paracompact spaces and continuous self-mappings. We consider geometric properties of spaces under the action of the subfunctor Pf of the functor P of probability measures and show that this functor Pf is an open σ-p.i.c. functor that preserves soft mappings and various types of topological spaces. [ABSTRACT FROM AUTHOR]

In the paper, the extremal boundary of the space of weakly additive, order-preserving, normalized, positive-homogeneous, and semiadditive functionals on a compact space is studied. The dimension of the extremal boundary of the convex compact OS(n) is found. [ABSTRACT FROM AUTHOR]

The present paper is an attempt to study the minimal BL-general fuzzy automata which realizes the given fuzzy behavior. Of two methods applied for construction of such automaton presented here, one has been based on Myhill- Nerode's theory while the other has been based on derivatives of the given fuzzy behavior. Meanwhile, the categories of BL-general fuzzy automata and fuzzy behavior, along with a functorial relationship between them, are introduced. [ABSTRACT FROM AUTHOR]

Concrete categories of functorial coalgebras are derived from given concrete categories under a certain commutativity condition satisfied by the underlying forgetful functor and endofunctors of its domain and codomain. When the base category is topological, so is that of functorial coalgebras when in addition to the commutativity condition the endofunctor of its domain preserves initial sources. We investigate the connection between fibres of objects in the topological category of coalgebras and those of the topological base category as well as some generalizations of the coalgebraic topological functor. [ABSTRACT FROM AUTHOR]