Search

In this paper, we present some characterizations of prime and maximal filters. Moreover, we introduce ∩-irreducible filters of an equality algebra, investigate some results about them and relations between maximal, prime, V-irreducible and ∩-irreducible filters in equality algebra. Also, we introduce spec- trum of an equality algebra and prove that the spectrum endowed with Zariski topology is a compact T0 topological space and maximal spectrum (as a subspace of that) is a compact T1 topological space. [ABSTRACT FROM AUTHOR]

In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [ABSTRACT FROM AUTHOR]

In the last decade, the Poincaré conjecture has probably been the most famous statement among all the contributions of Poincaré to the mathematics community. There have been many papers and books that describe various attempts and the final works of Perelman leading to a positive solution to the conjecture, but the evolution of Poincaré's works leading to this conjecture has not been carefully discussed or described, and some other historical aspects about it have not been addressed either. For example, one question is how it fits into the overall work of Poincaré in topology, and what are some other related questions that he had raised. Since Poincaré did not state the Poincaré conjecture as a conjecture but rather raised it as a question, one natural question is why he did this. In order to address these issues, in this paper, we examine Poincaré's works in topology in the framework of classifying manifolds through numerical and algebraic invariants. Consequently, we also provide a full history of the formulation of the Poincaré conjecture which is richer than what is usually described and accepted and hence gain a better understanding of overall works of Poincaré in topology. In addition, this analysis clarifies a puzzling question on the relation between Poincaré's stated motivations for topology and the Poincaré conjecture. [ABSTRACT FROM AUTHOR]

In the development of the theory of autometrized algebra, various types of research have been conducted. However, there are some properties like convex subalgebra, prime convex subalgebra, meet closed sets, regular convex subalgebra, and convex spectral topology on autometrized algebras that have not been studied yet. In this paper, we define the notions of convex subalgebras and congruence relations on an autometrized algebra. We demonstrate that the collection of all convex subalgebras of an autometrized algebra forms a lattice and distributive. In particular, we will show that there exists a one-to-one correspondence between the set of all convex subalgebras and the set of all congruences on an np-autometrized algebra. Furthermore, we explore prime convex subalgebras, meet closed subsets, and regular convex subalgebras and obtain some related results. For instance, we show that in a semiregular np-autometrized l-algebra, the intersection of a chain of prime convex subalgebra is a prime convex subalgebra. We also prove that any convex subalgebra in an autometrized algebra is the intersection of regular convex subalgebras. Lastly, we introduce the convex spectral topology of proper prime convex subalgebras in an autometrized l-algebra and discuss some fundamental facts. We also prove that a convex spectrum is compact in an np-autometrized l-algebra A if and only if A is generated by some element. Specifically, we demonstrate that the convex spectrum is a T1 - space and Hausdorff space. [ABSTRACT FROM AUTHOR]

With the untimely passing of Patrick Dehornoy in September 2019, the world of mathematics lost a brilliant scholar who made profound contributions to set theory, algebra, topology, and even computer science and cryptography. And I lost a dear friend and a strong influence in the direction of my own research in mathematics. In this paper, I will concentrate on his remarkable discovery that the braid groups are left-orderable, and its consequences, and its strong influence on my own research. I'll begin by describing how I learned of his work. [ABSTRACT FROM AUTHOR]

An S-Topological BE-Algebras(STBE-Algebras) is a BE-Algebra equipped with a special type of topology that makes the operation (defined on it)as S-topological continuous. In this paper, we discuss the separation axioms on a STBE-Algebras. [ABSTRACT FROM AUTHOR]

A Boolean algebra A equipped with a (finitely-additive) positive probability measure m can be turned into a metric space (A , d m) , where d m (a , b) = m ((a ∧ ¬ b) ∨ (¬ a ∧ b)) , for any a , b ∈ A , sometimes referred to as metric Boolean algebra. In this paper, we study under which conditions the space of atoms of a finite metric Boolean algebra can be isometrically embedded in R N (for a certain N) equipped with the Euclidean metric. In particular, we characterize the topology of the positive measures over a finite algebra A such that the metric space (At (A) , d m) embeds isometrically in R N (with the Euclidean metric). [ABSTRACT FROM AUTHOR]

This paper further studies topological structures induced by L-fuzzifying approximation operators, where L denotes a completely distributive De Morgan algebra. Firstly, the Alexandrov topologies induced by L-fuzzy relations are investigated with respect to L-fuzzifying approximation operators. Especially, the relationships among those Alexandrov topologies are discussed. Secondly, pseudo-similarity sets of L-fuzzy relations are proposed based on the Alexandrov topologies induced by upper and lower L-fuzzifying approximation operators. Meanwhile, the properties of pseudosimilarity sets are discussed, where some examples are presented to show the differences between similarity set of fuzzy relations and pseudo-similarity set of L-fuzzy relations. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to generalize the classical concept of a positive operator, and to develop a general extension theory, which overcomes not only the lack of a Hilbert space structure, but also the lack of a normable topology. The concept of anti‐duality carries an adequate structure to define positivity in a natural way, and is still general enough to cover numerous important areas where the Hilbert space theory cannot be applied. Our running example – illustrating the applicability of the general setting to spaces bearing poor geometrical features – comes from noncommutative integration theory. Namely, representable extension of linear functionals of involutive algebras will be governed by their induced operators. The main theorem, to which the vast majority of the results is built, gives a complete and constructive characterization of those operators that admit a continuous positive extension to the whole space. Various properties such as commutation, or minimality and maximality of special extensions will be studied in detail. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the statistically multiplicative convergent sequences in locally solid Riesz algebras with respect to algebra multiplication and solid topology. We study this concept and the notion of stm -bounded sequences, and also, we prove some relations between this convergence and the other convergences such as the order convergence and the statistical convergence in topological spaces. Also, we give some results related to semiprime f -algebras. [ABSTRACT FROM AUTHOR]