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We show that for a partial action η with closed domain of a compact group G on a profinite space X the space of orbits X/~G is profinite, this leads to the fact that when G is profinite the enveloping space XG is also profinite. Moreover, we provide conditions for the induced quotient map πG : X → X / ∼G of η to have a continuous section. Relations between continuous sections of πG and continuous sections of the quotient map induced by the enveloping action of η are also considered. At the end of this work, we prove that the category of actions on profinite spaces with countable number of clopen sets is reflective in the category of actions of compact Hausdorff spaces having countable number of clopen sets.

In this paper, we solve two problems concerning the ideally conjunctive join-semilattices. First, we show that L / R 1 (Id L) | L is ideally conjunctive for all join-semilattices L. Then we characterize those ideally conjunctive join-semilattices L such that coz a is compact for all a ∈ L. Moreover, we give the definition of conjunctive posets and prove that the category of ideally conjunctive join-semilattices and join homomorphisms is reflective in the category of conjunctive posets and weakly ideal-continuous maps. As a corollary, we obtain the free ideally conjunctive join-semilattices over conjunctive posets. [ABSTRACT FROM AUTHOR]

We show that for a nice partial action n with closed domain of a compact group G on a profinite space X the orbit space X/ ~G is profinite, this leads to the fact that when G is profinite the enveloping space XG is also profinite. Moreover, we provide conditions for the induced quotient map πG: X → ~G of η to have a continuous section. Relations between continuous sections of πG and continuous sections of the quotient map induced by the enveloping action of η are also considered. At the end of this work, we prove that the category of separately continuous actions on profinite spaces is reflective in the category of separately continuous actions on compact Hausdorff spaces. [ABSTRACT FROM AUTHOR]

In this paper, the authors introduce the notion of weak r-level LM-G-filter spaces and strong p-level LM-G-filter spaces and discuss certain properties of these spaces. The study identifies [Formula: see text]-G, the category of weak r-level LM-G-filter spaces as an isomorphism-closed bireflective full subcategory of LM-G, the category of LM-G-filter spaces. It is also proved that [Formula: see text]-G, the category of strong p-level LM-G-filter spaces is an isomorphism-closed bicoreflective full subcategory of LM-G. Moreover, level decompositions of LM-G-filter spaces are studied and some properties of the associated L-pre G-filter spaces are obtained.

This paper is devoted to the study of the T0 separation axiom for a neutrosophic topological space. We give a categorical study of these spaces in a special case of neutrosophic topological spaces called neutrosophic saturated topological spaces. [ABSTRACT FROM AUTHOR]

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 this paper, the authors introduce the notion of weak r-level LM-G-filter spaces and strong p-level LM-G-filter spaces and discuss certain properties of these spaces. The study identifies W L M r -G, the category of weak r-level LM-G-filter spaces as an isomorphism-closed bireflective full subcategory of LM-G, the category of LM-G-filter spaces. It is also proved that S L M p -G, the category of strong p-level LM-G-filter spaces is an isomorphism-closed bicoreflective full subcategory of LM-G. Moreover, level decompositions of LM-G-filter spaces are studied and some properties of the associated L-pre G-filter spaces are obtained. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to solve two open problems in topological residuated lattices. In this paper, given a linear topological residuated lattice, we find sufficient and necessary conditions under which the original topology coincides with the new one induced by a system of filters of the residuated lattice. This result will be valid for finite and infinite BL -algebras, which is a complete answer to the open problem proposed by Zahiri and Borzooei in 2016 and improves the incomplete answer given by Yang et al. in 2018. Moreover, using the topology induced by a system of filters of a residuated lattice, we prove that the set of all filters of the residuated lattice and the set of all corresponding zero-dimensional linear topological residuated lattices have the same cardinality. This result gives a negative answer to the open problem proposed by Yang and Zhang in 2019. Finally, we show that the category of residuated lattices is the reflective subcategory of the category of linear topological residuated lattices. [ABSTRACT FROM AUTHOR]

Fuzzy closure operators play a significant role in fuzzy order theory. This paper aims to further enrich and improve the study of fuzzy closure operators. Based on the work of Bělohlávek and Yao, we shall continue to study the relative properties of fuzzy closure operators. First, we shall consider the extensions of L-subsets via fuzzy closure operators. Then we give an application of fuzzy closure operators, that is, by fuzzy closure operators we shall prove that the category CFPos of complete fuzzy posets and their fuzzy-join preserving maps is a reflective full subcategory of FPost, where FPost denotes the category of fuzzy posets and their fuzzy-existing-join preserving maps. [ABSTRACT FROM AUTHOR]

Motivated by certain types of ideals in pointfree functions rings, we define what we call P-sublocales in completely regular frames. They are the closed sublocales that are interior to the zero-sublocales containing them. We call an element of a frame L that induces a P-sublocale a P-element, and denote by Pel (L) the set of all such elements. We show that if L is basically disconnected, then Pel (L) is a frame and, in fact, a dense sublocale of L. Ordered by inclusion, the set S p (L) of P-sublocales of L is a complete lattice, and, for basically disconnected L, S p (L) is a frame if and only if Pel (L) is the smallest dense sublocale of L. Furthermore, for basically disconnected L, S p (L) is a sublocale of the frame S c (L) consisting of joins of closed sublocales of L if and only if L is Boolean. For extremally disconnected L, iterating through the ordinals (taking intersections at limit ordinals) yields an ordinal sequence L ⊇ Pel (L) ⊇ Pel 2 (L) ⊇ ⋯ ⊇ Pel α (L) ⊇ Pel α + 1 (L) ⊇ ⋯ that stabilizes at an extremally disconnected P-frame, that we denote by Pel ∞ (L) . It turns out that Pel ∞ (L) is the reflection to L from extremally disconnected P-frames when morphisms are suitably restricted. [ABSTRACT FROM AUTHOR]

We show that for a field K and n ≥ 1 {n\geq 1} , the soberification 𝒮 ⁢ (𝔸 n ⁢ (K)) {\mathcal{S}(\mathbb{A}^{n}(K))} of the affine n-space 𝔸 n ⁢ (K) {\mathbb{A}^{n}(K)} over K is homeomorphic to its spectralization ℬ ⁢ 𝒮 ⁢ (𝔸 n ⁢ (K)) {\mathcal{BS}(\mathbb{A}^{n}(K))} , and it can be embedded into the spectrum Spec ⁢ (K ⁢ [ X 1 , ... , X n ]) {\mathrm{Spec}(K[X_{1},\dots,X_{n}])}. Moreover, if the field K is algebraically closed, then there are homeomorphisms 𝒮 ⁢ (𝔸 n ⁢ (K)) ≈ ℬ ⁢ 𝒮 ⁢ (𝔸 n ⁢ (K)) ≈ Spec ⁢ (K ⁢ [ X 1 , ... , X n ]) \mathcal{S}(\mathbb{A}^{n}(K))\approx\mathcal{BS}(\mathbb{A}^{n}(K))\approx% \mathrm{Spec}(K[X_{1},\dots,X_{n}]). We also show that for a space X, the subspace z ⁢ Spec ⁢ (C ⁢ (X)) ⊆ Spec ⁢ (C ⁢ (X)) z\mathrm{Spec}(C(X))\subseteq\mathrm{Spec}(C(X)) of prime z-ideals of the ring C ⁢ (X) {C(X)} of real-valued continuous functions on X is homeomorphic to the space z ⁢ 𝒮 ⁢ ℛ ⁢ (X) {z\mathcal{SR}(X)} of prime z-filters with an appropriate topology and there is a homeomorphism ℬ ⁢ 𝒮 ⁢ (X) ≈ z ⁢ Spec ⁢ (C ⁢ (X)) {\mathcal{BS}(X)\approx z\mathrm{Spec}(C(X))} provided X is perfectly normal. [ABSTRACT FROM AUTHOR]

In this paper, a new approach to fuzzy convergence theory in the framework of stratified L-topological spaces is provided. Firstly, the concept of stratified L-prefilter convergence structures is introduced and it is shown that the resulting category is a Cartesian closed topological category. Secondly, the relations between the category of stratified L-prefilter convergence spaces and the category of stratified L-topological spaces are studied and it is proved that the latter can be embedded in the former as a reflective subcategory. Finally, the relations between the category of stratified L-prefilter convergence spaces and the category of stratified L-Min convergence spaces (fuzzy convergence spaces in the sense of Min) are investigated and it is shown that the former can be embedded in the latter as a reflective subcategory. [ABSTRACT FROM AUTHOR]

We study the uniform connection properties of uniform local connectedness, a weaker variant of the latter, and a certain property S in the context of nearness frames. We show that the uniformly locally connected nearness frames form a reflective subcategory of the category of nearness frames whose underlying frame is locally connected. Amongst other results we show that these uniform connection properties are conserved and reflected by perfect nearness extensions which are uniformly regular. [ABSTRACT FROM AUTHOR]

In this paper by using the ring of real-valued continuous functions $C(X)$, we prove a theorem in profinite spaces which states that for a compact Hausdorff space $X$, the set of its connected components $X/_{\sim}$ endowed with the quotient topology is a profinite space. Then we apply this result to give an alternative proof to the fact that the category of profinite spaces is a reflective subcategory in the category of compact Hausdorff spaces. Finally, under some circumstances on a space $X$, we compute the connected components of the space $t(X)$ in terms of the ones of the space $X$.

In this paper, the concept of extensional L-fuzzy Q-convergence structures is proposed. It is shown that there is an adjunction between the category of extensional L-fuzzy topological spaces and the category of extensional L-fuzzy topological spaces. In particular, the category of extensional L-fuzzy topological spaces can be embedded in the category of extensional L-fuzzy Q-convergence spaces as a reflective subcategory. Also, the notion of topological extensional L-fuzzy Q-convergence spaces is introduced and the resulting category is shown to be isomorphic to the category of extensional L-fuzzy topological spaces. [ABSTRACT FROM AUTHOR]

Following the concept of topological theory of S.E. Rodabaugh, this paper introduces a new approach to (lattice-valued) bornology, which is based in bornological theories, and which is called variety-based bornology. In particular, motivated by the notion of topological system of S. Vickers, we introduce the concept of variety-based bornological system, and show that the category of variety-based bornological spaces is isomorphic to a full reflective subcategory of the category of variety-based bornological systems. [ABSTRACT FROM AUTHOR]

In this paper, based on a complete residuated lattice, we introduce the concept of fuzzy cut-stable maps and discuss the relationship between fuzzy cut-stable maps and some special maps. We also obtain the extension property of fuzzy cut-stable maps. Furthermore, it is proved that the category of fuzzy complete (continuous) lattices with fuzzy complete homomorphisms is a full reflective subcategory of the category of fuzzy (precontinuous) posets with fuzzy cut-stable maps. [ABSTRACT FROM AUTHOR]

Are all subcategories of locally finitely presentable categories that are closed under limits and ƛ-filtered colimits also locally presentable? For full subcategories the answer is affirmative. Makkai and Pitts proved that in the case ƛ = ... the answer is affirmative also for all iso-full subcategories, i. e., those containing with every pair of objects all isomorphisms between them. We discuss a possible generalization of this from ... to an arbitrary ƛ. [ABSTRACT FROM AUTHOR]

This paper extends the Day Reflection Theorem to skew monoidal categories. We also provide conditions under which a skew monoidal structure can be lifted to the category of Eilenberg-Moore algebras for a comonad. [ABSTRACT FROM AUTHOR]

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the L'evy hierarchy. For example, the statement that, for a class S of morphisms in a locally presentable category C of structures, the orthogonal class of objects is a small-orthogonality class (hence reflective) can be proved in ZFC if S is 61, while it follows from the existence of a proper class of supercompact cardinals if S is 62, and from the existence of a proper class of what we call C.n/-extendible cardinals if S is 6nC2 for n = 1. These cardinals form a new hierarchy, and we show that Vopenka's principle is equivalent to the existence of C.n/-extendible cardinals for all n. As a consequence of our approach, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, is implied by the existence of arbitrarily large supercompact cardinals. This follows from the fact that E'-equivalence classes are 62, where E denotes a spectrum treated as a parameter. In contrast with this fact, E'-equivalence classes are 61, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC. [ABSTRACT FROM AUTHOR]

Motivated by the concept of lattice-valued topological system of J.T. Denniston, A. Melton, and S.E. Rodabaugh, which extends lattice-valued topological spaces, this paper introduces the notion of lattice-valued bornological system as a generalization of lattice-valued bornological spaces of M. Abel and A. Šostak. We aim at (and make the first steps towards) the theory, which will provide a common setting for both lattice-valued point-set and point-free bornology. In particular, we show the algebraic structure of the latter. [ABSTRACT FROM AUTHOR]

Several methods for constructing left determined model structures are expounded. The starting point is Olschok's work on locally presentable categories. We give sufficient conditions to obtain left determined model structures on a full reflective subcategory, on a full coreflective subcategory and on a comma category. An application is given by constructing a left determined model structure on star-shaped weak transition systems. [ABSTRACT FROM AUTHOR]

Higher dimensional central extensions of groups were introduced by G. Janelidze as particular instances of the abstract notion of covering morphism from categorical Galois theory. More recently, the notion has been extended to and studied in arbitrary semi-abelian categories. Here, we further extend the scope to exact Mal'tsev categories and beyond. For this, we consider conditions on a Galois structure Γ = (ℂ, 핏, I, H, η, ℰ) which insure the existence of an induced Galois structure Γ = (ℂ, 핏, I, H, η, ℰ) such that ℂ and 핏 are full subcategories of the arrow category (ℂ) consisting, respectively, of all morphisms in the class ℰ, and of all covering morphisms with respect to Γ. Moreover, we prove that Γ satisfies the same conditions as Γ, so that, inductively, we obtain, for each n ≥ 1, a Galois structure Γ = ( Γ), whose coverings we call n + 1-fold central extensions. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to extend the dualizing object approach to Stone duality to the noncommutative setting of skew Boolean algebras. This continues the study of noncommutative generalizations of different forms of Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson, Lawson and Lenz, Resende, and also the current author. In this paper we construct a series of dual adjunctions between the categories of left-handed skew Boolean algebras and Boolean spaces, the unital versions of which are induced by dualizing objects $\{ 0, 1, \ldots , n+ 1\} $, $n\geq 0$. We describe the categories of Eilenberg-Moore algebras of the monads of the adjunctions and construct easily understood noncommutative reflections of left-handed skew Boolean algebras, where the latter can be faithfully embedded (if $n\geq 1$) in a canonical way. As an application, we answer the question that arose in a recent paper by Leech and Spinks to describe the left adjoint to their ‘twisted product’ functor $\omega $. [ABSTRACT FROM AUTHOR]

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We characterize abstract Morita contexts in several bicategories. In particular, we use heteromorphisms for the bicategory $\mathbb{Prof} $ of categories and profunctors and coreflective subcategories for $\mathbb {Cat}$ (categories and functors). In addition, we prove general statements concerning strict Morita contexts, and we give new equivalent forms to the standard notions of adjointness, category equivalence and Morita equivalence by studying the collage of a profunctor. [ABSTRACT FROM AUTHOR]

Abstract: Following John Kennison, a flow (or discrete dynamical system) in a category C is a couple , where X is an object of C and is a morphism, called the iterator. If and are flows in C, then is a morphism of flows from to if . We let denote the resulting category of flows in C. This paper deals with and , where Set and Top denote respectively the categories of sets and topological spaces. By a Gottschalk flow, we mean a flow in Top satisfying the following conditions: [(i)] If is any almost periodic point of f, then the closure is a minimal set of f; [(ii)] All points in any minimal set of f are almost periodic points. As proven by Gottschalk, if X is a compact Hausdorff space and is a continuous function, then is a Gottschalk flow. In this paper, we prove that for any flow of Set, there is a topology on X for which is a Gottschalk flow in Top. This, actually, defines a covariant functor from into . The main result of this paper provides a characterization of spaces in the image of the functor in order-theoretical terms. Some categorical properties of and are also given. [Copyright &y& Elsevier]

Abstract: We investigate a Galois connection in poset enriched categories between subcategories and classes of morphisms, given by means of the concept of right-Kan injectivity, and, specially, we study its relationship with a certain kind of subcategories, the KZ-reflective subcategories. A number of well-known properties concerning orthogonality and full reflectivity can be seen as a particular case of the ones of right-Kan injectivity and KZ-reflectivity. On the other hand, many examples of injectivity in poset enriched categories encountered in the literature are closely related to the above connection. We give several examples and show that some known subcategories of the category of -topological spaces are right-Kan injective hulls of a finite subcategory. [Copyright &y& Elsevier]

Abstract: The paper provides variety-based extensions of the concepts of (lattice-valued) interchange system and space, introduced by Denniston, Melton and Rodabaugh, and shows that variety-based interchange systems incorporate topological systems of Vickers, state property systems of Aerts, Chu spaces (over the category of sets in the sense of Pratt) of P.-H. Chu and contexts (of formal concept analysis) of Wille. The paper also provides an explicit description of (co)limits in the category of variety-based topological systems and applies the obtained results to extend the claim of Denniston et al. that the category of topological systems of Vickers is small initially topological over the category of sets. [Copyright &y& Elsevier]

An open subset U of a topological space X is called intersection compact open, or ICO, if for every compact open set Q of X, U ∩ Q is compact. A continuous map f of topological spaces will be called spectral if f-1 carries ICO sets to ICO sets. Call a topological space Xhemispectral, if the intersection of two ICO sets of X is an ICO. Let HSPEC be the category whose objects are hemispectral spaces and arrows spectral maps. Let SPEC be the full subcategory of HSPEC whose objects are spectral spaces. The main result of this paper proves that SPEC is a reflective subcategory of HSPEC. This gives a complete answer to Problem BST1 of "O. Echi, H. Marzougui and E. Salhi, Problems from the Bizerte-Sfax-Tunis seminar, in Open Problems in Topology II, ed. E. Pearl (Elsevier, 2007), pp. 669-674." [ABSTRACT FROM AUTHOR]

We characterize nearness frames whose completions are fine (we call them quotient-fine), and show that the subcategory QfNFrm they form is reflective in the category of strong nearness frames. The resulting functor commutes with the completion functor. QfNFrm is isomorphic to the subcategory of the functor category ( RegFrm) given by the dense onto $h\colon M\to L$, where 2 denotes the category with only two objects and exactly one morphism between them. [ABSTRACT FROM AUTHOR]

We study topological spaces with a distinguished set of paths, called directed paths. Since these directed paths are generally not reversible, the directed homotopy classes of directed paths do not assemble into a groupoid, and there is no direct analog of the fundamental group. However, they do assemble into a category, called the fundamental category. We define models of the fundamental category, such as the fundamental bipartite graph, and minimal extremal models which are shown to generalize the fundamental group. In addition, we prove van Kampen theorems for subcategories, retracts, and models of the fundamental category. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop a homotopical algebra for graphs, relevant to the zeta series and the spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equivalences for this model structure are the Acyclics (graph morphisms which preserve cycles). The cofibrations and fibrations for the model are determined from the class of Whiskerings (graph morphisms produced by grafting trees). Our model structure seems to fit well with the importance of acyclic directed graphs in many applications. [ABSTRACT FROM AUTHOR]

Let frA denote the category of f-rings which are reduced and Archimedean, and let Φ be the (nonfull) subcategory of such rings with identity (each with the natural morphisms). Some time ago, the second author showed, using his representation theory, that for each A ∈ | frA| there is a certain minimal embedding uA:A→ uA ∈ | Φ|. More recently, he has revisited the representation theory, expanding it to include the representation of morphisms. Based upon this, the present article analyzes the operator u:| frA| → Φ: the construction of uA is tidied, several characterizations of the pair (uA, uA) are given, and the relation between the maximal ideal structures of A and uA is described. Membership in the class U of frA-morphisms that are "u-extendable" is characterized and it is shown that U = (| frA|,U) is a category in which Φ is a full essentially-reflective subcategory. The frA-objects are characterized for which, respectively, ∀ B(frA(A, B) = U (A, B)), and, ∀ B ≠ 0(frA(B, A) = U(B, A)). [ABSTRACT FROM AUTHOR]

The category-theoretic nature of general frames for modal logic is explored. A new notion of "modal map" between frames is defined, generalizing the usual notion of bounded morphism/p-morphism. The category Fm of all frames and modal maps has reflective subcategories CHFm of compact Hausdorff frames, DFm of descriptive frames, and UEFm of ultrafilter enlargements of frames. All three subcategories are equivalent, and are dual to the category of modal algebras and their homomorphisms. An important example of a modal map that is typically not a bounded morphism is the natural insertion of a frame A into its ultrafilter enlargement EA. This map is used to show that EA is the free compact Hausdorff frame generated by A relative to Fm. The monad E of the resulting adjunction is examined and its Eilenberg-Moore category is shown to be isomorphic to CHFm. A categorical equivalence between the Kleisli category of E and UEFm is defined from a construction that assigns to each frame A a frame A* that is "image-closed" in the sense that every point-image {b : aRb} in A is topologically closed. A* is the unique image-closed frame having the same ultrafilter enlargement as A. These ideas are connected to a category 2U shown by S. K. Thomason to be dual to the category of complete and atomic modal algebras and their homomorphisms. 2U is the full subcategory of the Kleisli category of E based on the Kripke frames. [ABSTRACT FROM AUTHOR]

If (R, M) and (S, N) are quasi-local domains and f : R → S is a ring homomorphism, then f is said to be a strong local homomorphism if f(M) = N. Let PVD be the category whose objects are the pseudo-valuation domains that are not fields and whose morphisms are the strong local homomorphisms; let VD be the full subcategory of PVD whose objects are all the valuation domains that are not fields. Then VD is shown to be a reflective subcategory of PVD. This result is extended by obtaining a reflective conclusion for a category whose class of objects properly contains all pseudo-valuation domains. [ABSTRACT FROM AUTHOR]

Let L be a meet continuous lattice. It is proved that the category Top of topological spaces can be embedded in the category of stratified L-topological spaces as a concretely both reflective and coreflective full subcategory if and only if L is a continuous lattice. [ABSTRACT FROM AUTHOR]