  In this paper, we enlarge the language of MTL-algebras by a unary operation
$\forall$ equationally described so as to abstract algebraic properties of the
universal quantifier "for any" in its original meaning. The resulting class of
algebras will be called \emph{MTL-algebras with universal quantifiers}
(UMTL-algebras for short). After discussing some basic algebraic properties of
UMTL-algebras, we start a systematic study of the main subclasses of
UMTL-algebras, some of which constitute well known algebras: UMV-algebras and
monadic Boolean algebra. Then we give some characterizations of representable,
simple, semsimple UMTL-algebras, and obtain some representations of
UMTL-algebras. Finally, we establish modal monoidal t-norm based logic and
prove that is completeness with respect to the variety of UMTL-algebras, and
then obtain that a necessary and sufficient condition for the modal monoidal
t-norm based logic to be semilinear.
