Solutions to two open problems in topological residuated lattices.

The main aim of this paper is to solve two open problems in topological residuated lattices. In this paper, given a linear topological residuated lattice, we find sufficient and necessary conditions under which the original topology coincides with the new one induced by a system of filters of the residuated lattice. This result will be valid for finite and infinite BL -algebras, which is a complete answer to the open problem proposed by Zahiri and Borzooei in 2016 and improves the incomplete answer given by Yang et al. in 2018. Moreover, using the topology induced by a system of filters of a residuated lattice, we prove that the set of all filters of the residuated lattice and the set of all corresponding zero-dimensional linear topological residuated lattices have the same cardinality. This result gives a negative answer to the open problem proposed by Yang and Zhang in 2019. Finally, we show that the category of residuated lattices is the reflective subcategory of the category of linear topological residuated lattices. [ABSTRACT FROM AUTHOR]