School of Information Environment, Tokyo Denki University, P.O. Box 270-1382, Inzai, Japan
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 $X$, $bigcap {P_{lambda} ,|,P_{lambda} {rm is an} m{rm -filter} } = {1}$ 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).
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