Representation of Prelinear Residuated Algebras

In 1998, Hájek established a representation theorem of BL-algebrs as all subdirect products of linear BL-algebrs. We establish a similar result for the much wider class of prelinear residuated algebras, in which neither the lattice structure nor the divisibility of the monoid operation is assumed. We show, in the case of prelinear residuated lattices, that this order embedding becomes a lattice embedding. In consequence, prelinear residuated lattices inherit several properties from products of residuated chains. We provide alternative proofs for some of those properties, which avoid the use of the axiom of choice. Copyright c © 2007 Yang’s Scientific Research Institute, LLC. All rights reserved.