A topology induced by uniformity on BL-algebras

BL-algebras have been invented by P. Hájek [3] in order to provide an algebraic proof of the completeness theorem of “Basic Logic” (BL, for short) arising from the continuous triangular norms, familiar in the fuzzy logic framework. The above notion is generalized to an algebraic system in which the required conditions are fulfilled (Definition 2.1). Filters in BL-algebras are also defined. A filter is a special subset of the BL-algebra that contains elements somehow “close” to 1, which is called the largest element. This leads us to think of a neighborhood system in a BL-algebra, collecting those subsets whose elements are “close” to each other. Some researchers [1, 2, 3] choose to have equivalence or congruence classes of some equivalence or congruence relation with respect to a given filter, respectively. In this paper, we consider a collection of filters and use congruence relation with respect to filters to define a uniformity and turn the BL-algebra into a uniform topological space with the desired subset as the open sets. Towards our goal, we renew some needed algebraic notions in Section 2. Then we consider the uniformity based on congruence relations with respect to a given collection of filters and construct the induced topology by this uniformity in Section 3. In the last sections we study the properties of these topologies such as compactness regarding different filters. Our future research will be investigating the effect of these topologies when we translate their meaning into the related BL-logic.