A note on some types of filters in MTL-algebras

Borzooei et al. recently introduced the notion of associative filters in MTL-algebras and studied some of their properties. In this paper we prove that the only proper filter of MTL-algebra L that is associative according to their definition is L ∖ { 0 } . In fact, in any MTL-algebra L if there exists a proper associative filter F in L then F = L ∖ { 0 } .