In this note, we introduce the concept of a fuzzy filter of a BL- algebra, with respect to a t-norm briefly, T-fuzzy filters, and give some related results. In particular, we prove Representation Theorem in BL-algebras. Then we generalize the notion of a fuzzy congruence (in a BL-algebra) was defined by Lianzhen et al. to a new fuzzy congruence, specially with respect to a t- norm. We prove that there is a correspondence bijection between the set of all T-fuzzy filters of a BL-algebra and the set of all T-fuzzy congruences in that BL-algebra. Next, we show how T-fuzzy filters induce T-fuzzy congruences, and construct a new BL-algebras, called quotient BL-algebras, and give some homomorphism theorem.

In last ten years many mathematicians have studied properties of BL-algebras endowed with a topology. For example A. Di Nola and L. Leustean [5] studied compact representations of BL-algebras, L. C. Ciungu [4] investigated some concepts of convergence in the class of perfect BL-algebras, J. Mi Ko and Y. C. Kim [13] studied relationships between closure operators and BL-algebras, M. Haveshki, E. Eslami and A. Broumand Saeid [9] applied filters to construct a topology on BL-algebras. In this paper we define semitopological and topological BLalgebras and derive here conditions that imply a BL-algebra to be a semitopological or topological BL-algebra.

In this paper, we generalize the concepts of para and quasi topological MV-algebras, which was first introduced by najafi et al. in 2017, to BL-algebras as para and quasi topological BL-algebras and elaborate these concepts via some examples. We further derive and prove some theorems by employing pre-filters and a fundamental system of neighborhoods.

In this paper, by considering the notion of L-algebra, we show that there are relations between L-algebras and some of other logical algebras such as residuated lattices, MTL-algebras, BL-algebras, MV-algebras, BCK-algebras, equality algebras, EQ-algebras and Hilbert algebras. The aim of this paper is to find under what conditions L-algebras are equivalent to these logical algebras.

In this paper, we introduce the notion of multiplier in BL-algebra and study relationships between multipliers and some special mappings, likeness closure operators, homomorphisms and (ʘ, Ú)-derivations in BL-algebras. We introduce the concept of idempotent multipliers in BL-algebra and weak congruence and obtain an interconnection between idempotent multipliers and weak congruences. Also, we introduce the special multiplier ap and study some properties. Finally, we show that if A is a boolean algebra, then the set of all multipliers of A is a BL-algebra under some conditions.