Fuzzy ee-regular spaces and strongly ee-irresolute mappings

The aim of this paper is to introduce fuzzy ($e$, almost) $e^{*}$-regular spaces in $check{S}$ostak's fuzzy topological spaces. Using the $r$-fuzzy $e$-closed sets, we define $r$-($r$-$theta$-, $r$-$etheta$-) $e$-cluster points and their properties. Moreover, we investigate the relations among $r$-($r$-$theta$-, $r$-$etheta$-) $e$-cluster points, $r$-fuzzy ($e$, almost) $e^{*}$-regular spaces and their functions

Relational Approach to the L-Fuzzy Concept Analysis

Modern industrial production systems benefit from the classification and processing of objects and their attributes. In general, the object classification procedure can coincide with vagueness. Vagueness is a common problem in object analysis that exists at various stages of classification, including ambiguity in input data, overlapping boundaries between classes or regions, and uncertainty in defining or extracting the properties and relationships of objects. To manage the ambiguity mentioned in the classification of objects, using a framework for L-fuzzy relations, and displaying such uncertainties by it can be a solution. Obtaining the least unreliable and uncertain output associated with the original data is the main concern of this thesis. Therefore, my general approach to this research can be categorized as follows: We developed an L-Fuzzy Concept Analysis as a generalization of a regular Concept Analysis. We start our work by providing the input data. Data is stored in a table (database). The next step is the creation of the contexts and concepts from the given original data using some structures. In the next stage, rules, or patterns (Attribute Implications) from the data will be generated. This includes all rules and a minimal base of rules. All of them are using L-fuzziness due to uncertainty. This requires L-fuzzy relations that will be implemented as L -valued matrices. In the end, everything is nicely packed in a convenient application and implemented in Java programming language. Generally, our approach is done in an algebraic framework that covers both regular and L -Fuzzy FCA, simultaneously. The tables we started with are already L-valued (not crisp) in our implementation. In other words, we work with the L-Fuzzy data directly. This is the idea here. We start with vague data. In simple terms, the data is shown using L -valued tables (vague data) trying to relate objects with their attributes at the start of the implementation. Generating attribute implications from many-valued contexts by a relational theory is the purpose of this thesis, i.e, a range of degrees is used to indicate the relationship between objects and their properties. The smallest degree corresponds to the classical no and the greatest degree corresponds to the classical yes in the table

Concepts of vertex regularity in cubic fuzzy graph structures with an application

The cubic fuzzy graph structure, as a combination of cubic fuzzy graphs and fuzzy graph structures, shows better capabilities in solving complex problems, especially in cases where there are multiple relationships. The quality and method of determining the degree of vertices in this type of fuzzy graphs simultaneously supports fuzzy membership and interval-valued fuzzy membership, in addition to the multiplicity of relations, motivated us to conduct a study on the regularity of cubic fuzzy graph structures. In this context, the concepts of vertex regularity and total vertex regularity have been informed and some of its properties have been studied. In this regard, a comparative study between vertex regular and total vertex regular cubic fuzzy graph structure has been carried out and the necessary and sufficient conditions have been provided. These degrees can be easily compared in the form of a cubic number expressed. It has been found that the condition of the membership function is effective in the quality of degree calculation. In the end, an application of the degree of vertices in the cubic fuzzy graph structure is presented

The moduli space of matroids

In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are compatible with the respective matroid theories. Moreover, fuzzy rings are ordered blueprints and lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection pastures. In the second part, we construct moduli spaces for matroids over pastures. We show that, for any non-empty finite set $E$, the functor taking a pasture $F$ to the set of isomorphism classes of rank-$r$ $F$-matroids on $E$ is representable by an ordered blue scheme $Mat(r,E)$, the moduli space of rank-$r$ matroids on $E$. In the third part, we draw conclusions on matroid theory. A classical rank-$r$ matroid $M$ on $E$ corresponds to a $\mathbb{K}$-valued point of $Mat(r,E)$ where $\mathbb{K}$ is the Krasner hyperfield. Such a point defines a residue pasture $k_M$, which we call the universal pasture of $M$. We show that for every pasture $F$, morphisms $k_M\to F$ are canonically in bijection with $F$-matroid structures on $M$. An analogous weak universal pasture $k_M^w$ classifies weak $F$-matroid structures on $M$. The unit group of $k_M^w$ can be canonically identified with the Tutte group of $M$. We call the sub-pasture $k_M^f$ of $k_M^w$ generated by ``cross-ratios' the foundation of $M$,. It parametrizes rescaling classes of weak $F$-matroid structures on $M$, and its unit group is coincides with the inner Tutte group of $M$. We show that a matroid $M$ is regular if and only if its foundation is the regular partial field, and a non-regular matroid $M$ is binary if and only if its foundation is the field with two elements. This yields a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.Comment: 83 page