A metric-like topology yon BL-algebras

Abstract

This paper is devoted to introduce a topology on BL-algebras, makes them semitopological algebras. For any BL-algebra $\mathcal{L}=(L,\wedge ,\vee ,\ast ,\to ,0,1)$, the introduced topology is defined by a distance-like function between elements of $L$ which is defined by $a\leftrightarrow b=(a\to b)\ast (b\to a)$. 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.