Lattice-valued Topology Research Articles

Monoidal closedness of the category of $\top$-semiuniform convergence spaces

Lattice-valued semiuniform convergence structures are important mathematical structures in the theory of lattice-valued topology. Choosing a complete residuated lattice $L$ as the lattice background, we introduce a new type of lattice-valued filters using the tensor and implication operations on $L$, which is called $\top$-filters. By means of $\top$-filters, we propose the concept of $\top$-semiuniform convergence structures as a new lattice-valued counterpart of semiuniform convergence structures. Different from the usual discussions on lattice-valued semiuniform convergence structures, we show that the category of $\top$-semiuniform convergence spaces is a topological and monoidal closed category when $L$ is a complete residuated lattice without any other requirements.

Characterizations of Pointwise Pseudometrics via Pointwise Closed-Ball Systems

Pointwise pseudo-quasi-metrics play an important role in the theory of lattice-valued topology (L-topology). Bearing in mind that closed balls and their relations with pseudo-quasi-metrics have historically attracted the attention of mathematicians, it is very surprising that no attention has been paid to the relations between pointwise pseudo-quasi-metrics and closed balls. In this paper, we firstly introduce the concept of pointwise closed-ball systems and prove that the resulting category is isomorphic to that of pointwise pseudo-quasi-metrics. Subsequently, we study the topological properties of pointwise pseudo-quasi metrics via pointwise closed-ball systems. Interestingly, the <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex>$L$</tex> </formula> -topologies defined by open sets and complements of closed sets coincide for any pointwise pseudo-metric. Finally, we expose an important theoretical application of the pointwise closed-systems in providing a different and relatively simpler proof of the celebrated metrization theorem of the L-fuzzy real line.

Weak hyper semi-quantales and weak hypervalued topological spaces

The purpose of this paper is to construct a weak hyper semi-quantale as a generalization of the concept of semi-quantale and used it as an appropriate hyperlattice-theoretic basis to formulate new lattice-valued topological theories. Based on such weak hyper semi-quantale, we aim to construct the notion of a weak hypervalued-topology as a generalized form of the so-called lattice-valued topology (or many-valued topology). Some properties of weak hyper semi-quantales and weak hypervalued-topologies will be studied. An adjunction between the category of weak hyper semi-quantales and the category of weak hypervalued quasi-topological spaces will be established.

On the quantale of quantic nuclei

It is well known that the set $$\textit{NL}$$ of all nuclei on a frame L is again a frame with the pointwise order. As a non-commutative generalization of frames, quantales were introduced by Mulvey in 1986. Niefield and Rosenthal attempted to define a binary operation & on the set $$\textit{NQ}$$ of all quantic nuclei of a quantale Q such that ($$\textit{NQ}$$, &) is again a quantale. However, Sun showed that the way of Niefield and Rosenthal failed for a general quantale. Girard quantales are a special class of quantales, which play an important role in the study of linear intuitionistic logic, quantitative domain, lattice-valued topology and enriched category. In this note, we give an example to indicate that $$\textit{NQ}$$, together with & defined by Niefield and Rosenthal, is in general not a quantale even for a Girard quantale Q. However, with respect to & on a Girard quantale Q, we redefine a binary operation $$\odot $$ on $$\textit{NQ}$$ such that $$(\textit{NQ}^{op}, \odot )$$ is a quantale.

Categorically-algebraic topology and its applications

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respective categories of topological structures are topological over their ground categories. The theory also extends the notion of topological system of S.~Vickers (and its numerous many-valued modifications of J.~T.~Denniston, A.~Melton and S.~E.~Rodabaugh), and shows that the categories of catalg topological structures are isomorphic to coreflective subcategories of the categories of catalg topological systems. This extension initiates a new approach to soft topology, induced by the concept of soft set of D.~Molodtsov, and currently pursued by various researchers.

Lattice-valued preuniform convergence spaces

Using the intrinsic lattice-valued inclusion orders of L-subsets and of stratified L-filters, the concept of a stratified L-preuniform convergence structure is proposed. This convergence structure is asymmetric and can be used to establish a framework of asymmetric lattice-valued space structures. The category of stratified L-preuniform convergence spaces is introduced and some categorical properties are presented. A reflective subcategory of stratified L-preuniform convergence spaces is found that is categorically isomorphic to strong stratified L-convergence spaces (originally called stratified L-ordered convergence spaces). Several subcategories of stratified L-preuniform convergence spaces are established and relations between the different subcategories are presented. We conclude that stratified L-preuniform convergence structures introduced here could play a role in the framework of lattice-valued asymmetric space structures in lattice-valued topology.

Interweaving algebra and topology: Lattice-valued topological systems

This paper is primarily dedicated to understanding the natural role that topological systems and lattice-valued topological systems play in understanding the relationship between algebra and topology, a relationship expressed by an “interweaving” of embeddings of (essentially) algebraic and topological categories. This natural role is revealed by how various categories for systems relate to locales, topological spaces, and various categories for lattice-valued topology.

Categorical foundations of variety-based topology and topological systems

The paper considers a new approach to fuzzy topology based on the concept of variety and developed in the framework of topological theories resembling those of Rodabaugh. As a result, a categorical generalization of the notion of topological system of Vickers is obtained, and its theory unfolded, which clarifies the relations between algebra and topology. We also justify the use of semi-quantales as the basic underlying structure for doing lattice-valued topology as well as provide a categorical framework incorporating the theory of bitopological spaces.

Necessity of non-stratified and anti-stratified spaces in lattice-valued topology

This paper surveys the necessity of non-stratified spaces from these viewpoints: the characteristic functor, the L-spectrum and L-soberification functors, the upper free functor associated with L-valued frames, and two functorial embeddings associated with topological systems from semantic domains. Interestingly, additional arguments also emerge en route for stratified spaces, so that this paper ultimately argues for the necessity of both stratified and non-stratified spaces in fixed-basis topology (the schemum of L- Top's) and variable-basis topology (e.g., Loc- Top).

Functorial relationships between lattice-valued topology and topological systems

This paper investigates functorial relationships between lattice-valued topology (arising from fuzzy sets and fuzzy logic) and topological systems (arising from topological and localic aspects of domains and finite observational logic in computer science). Two such relationships are embeddings from TopSys into Loc-Top, both having two fold significance: for computer science the significance is that TopSys is not topological over Set × Loc, yet Loc-Top is topological over Set × Loc; hence these embeddings can be used to construct in Loc-Top the unique initial [final] lifts of all forgetful functor structured sources [sinks] in TopSys; and for topology, the significance is that both embeddings generate anti-stratified topological spaces from ordinary topological spaces and spatial locales rewritten as topological systems, thus justifying the current structural axioms of Loc-Top and lattice-valued topology (which include all anti-stratified, non-stratified, and stratified spaces).

Functorial comparisons of bitopology with topology and the case for redundancy of bitopology in lattice-valued mathematics

This paper studies various functors between (lattice-valued) topology and (lattice-valued) bitopology, including the expected “doubling” functor E d : L-Top → L-BiTop and the “cross” functor E× : L-BiTop → L 2 -Top introduced in this paper, both of which are extremely well-behaved strict, concrete, full embeddings. Given the greater simplicity of lattice-valued topology vis-a-vis lattice-valued bitopology and the fact that the class of L2-Top’s is strictly smaller than the class of L-Top’s encompassing fixed-basis topology, the class of E×’s makes the case that lattice-valued bitopology is categorically redundant. As a special application, traditional bitopology as represented by BiTop is (isomorphic in an extremely well-behaved way to) a strict subcategory of 4-Top, where 4 is the four element Boolean algebra; this makes the case that traditional bitopology is a special case of a much simpler fixed-basis topology.

Category theoretic aspects of chain-valued frames: Part II: Applications to lattice-valued topology

This paper is Part II of a two-part series dealing with category theoretic aspects of chain-valued frames. Using the categorical properties established for L- Frm in Part I for L a complete chain, this paper constructs “upper” free functor L and “lower” free functor R . The functor L is used to create a class of non-generated L-frames, factor the LPT spectrum functor through the Σ spectrum functor, resolve the relationship between L-sobriety and ι L -sobriety, and give insight into the construction of “universal” L-topological spaces; and the functor R is used to create a class of non-generated L-frames, factor the ω L functor through a new spectrum functor Σ * , and give insight into the construction of “co-universal” L-topological spaces. These facts—universal spaces are anti-stratified, co-universal spaces are stratified, L- Frm produces both kinds of spaces via L and R , L- Frm (via its dual L- Loc) is adjunctive with L- Top—comprise a coherent argument that L- Top must accommodate both kinds of spaces, resolving the philosophical debate over the place of the constant maps condition with respect to the axioms of L-topologies.

Category theoretic aspects of chain-valued frames: Part I: Categorical and presheaf theoretic foundations

This paper is Part I of a two-part series dealing with category theoretic aspects of chain-valued frames. A new categorical motivation for lattice-valued frames is given from presheaves, and then, under the assumption that L be a complete chain, it is established that standard category–theoretic properties (completeness, cocompleteness, factorization structures) hold for L - Frm (the category of L -frames and L -frame morphisms), properties which are a platform for the “upper” free functor L and “lower” free functor R used in Part II to give a broad range of applications of chain-valued frames to lattice-valued topology.

Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics

This paper deals with a broad question—to what extent is topology algebraic—using two specific questions: (1) what are the algebraic conditions on the underlying membership lattices which insure that categories for topology and fuzzy topology are indeed topological categories; and (2) what are the algebraic conditions which insure that algebraic theories in the sense of Manes are a foundation for the powerset theories generating topological categories for topology and fuzzy topology? This paper answers the first question by generalizing the Höhle-Šostak foundations for fixed-basis lattice-valued topology and the Rodabaugh foundations for variable-basis lattice-valued topology using semi-quantales; and it answers the second question by giving necessary and sufficient conditions under which certain theories—the very ones generating powerset theories generating (fuzzy) topological theories in the sense of this paper—are algebraic theories, and these conditions use unital quantales. The algebraic conditions answering the second question are much stronger than those answering the first question. The syntactic benefits of having an algebraic theory as a foundation for the powerset theory underlying a (fuzzy) topological theory are explored; the relationship between these two specific questions is discussed; the role of pseudo-adjoints is identified in variable-basis powerset theories which are algebraically generated; the relationships between topological theories in the sense of Adámek-Herrlich-Strecker and topological theories in the sense of this paper are fully resolved; lower-image operators introduced for fixed-basis mathematics are completely described in terms of standard image operators; certain algebraic theories are given which determine powerset theories determining a new class of variable-basis categories for topology and fuzzy topology using new preimage operators; and the theories of this paper are undergirded throughout by several extensive inventories of examples.