A metric-like topology yon BL-algebras

TL;DR

This paper introduces a topology on BL-algebras using a distance-like function, making them semitopological algebras, and explores its properties especially when applied to the continuous scale [0,1].

Contribution

It defines a new topology on BL-algebras based on a metric-like function and studies its properties, extending to dual algebras, which is a novel approach.

Findings

Topology admits properties similar to metric topology on [0,1] BL-algebra.

The topology can be extended to duals of BL-algebras.

Provides a framework for semitopological BL-algebras.

Abstract

This paper is devoted to introduce a topology on BL-algebras, makes them semitopological algebras. For any BL-algebra $\mathcal{L}=(L, \wedge, \vee, *, \to , 0, 1)$, the introduced topology is defined by a distance-like function between elements of $L$ which is defined by $a \leftrightarrow b=(a\to b)*(b\to a)$. We will show that when the continuous scale $[0,1]$ is endowed to be a BL-algebra, then this topology admits some of the most important properties of the metric topology. Finally, we will show that this topology can be examined by a similar topology on dual of BL-algebras as well.