On residuated lattices with universal quantifiers

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, bigcapP lambda, |,P lambda rmisanmrm−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).