A note on a method for constructing uniform topologies on algebras associated with logical systems

Article type: Research Article

Authors: Asadzadeh, Mohammad Sina | Rezaei, Gholam Reza^*; | Jamalzadeh, Javad

Affiliations: Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

Correspondence: [*] Corresponding author. Gholam Reza Rezaei, Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran. E-mail: grezaei@math.usb.ac.ir.

Abstract: In the recent years, many authors have used a single method for equipping algebraic structures with uniformities which are induced by families of algebraic objects. This paper is devoted to a description of this well-known method in general, and provides insight into those results which are obtained using the method. In fact, we prove that the uniform topology induced by this method coincides with a partition topology generated by an equivalence relation, and illustrate the logic behind the continuity of algebraic operations in these kinds of uniform topologies. Furthermore, the main topological properties of the partition topology induced by a congruence relation are presented. As an application, we explain why many results obtained from this method are trivial. These results have been collected from the works of several mathematicians on more than twenty different algebraic systems over the course of two decades.

Keywords: Algebraic structure, Bl-algebra, uniform structure, partition space, partition uniformity

DOI: 10.3233/JIFS-191709

Journal: Journal of Intelligent & Fuzzy Systems, vol. 39, no. 1, pp. 789-793, 2020