Abstract :
We consider properties of residuated lattices with universal quantifier and show that, for a residuated lattice $X$, $(X, \forall)$ is a residuated lattice with a quantifier if and only if there is an $m$-relatively complete substructure of $X$. We also show that, for a strong residuated lattice $X$, $\bigcap \{P_{\lambda}, P_{\lambda} is an m-filter ={1}$ and hence that any strong residuated lattice is a subdirect product of a strong residuated lattice with a universal quantifier $\{ X/P_{\lambda} \}$, where $P_{\lambda}$ is a prime $m$-filter. As a corollary of this result, we prove that every strong monadic MTL-algebra (BL- and MV-algebra) is a subdirect product of linearly ordered strong monadic MTL-algebras (BL- and MV-algebras, respectively).
