A Representation Theorem, Based on Ideals, for Stonean Residuated Lattices
Liviu-Constantin Holdon · 2020
Stonean residuated lattices form a variety which includes among its most important subvarieties the variety of Boolean algebras, Gödel algebras, product algebras and pseudo-complemented MTL-algebras (in fact Stonean residuated lattices are pseudocomplemented). In the present work, in the same spirit of [5], [15], we continue to study the lattice of ideals of a Stonean residuated lattice, we state the pseudocomplementedness theorem of the ideal lattice, the lattice of ideals is a complete distributive Browerian lattice and the representation theorem: Given an Stonean residuated lattice L the correspondence φ :I(L) →I(B(L)) that sends each ideal I ∈ I(L) to the ideal φ(I) = {x** : x ∈ I} of B(L) is a lattice isomorphism. In the light of our results the study of ideals in Stonean residuated lattices reduced to the study of ideals of the Boolean skeleton and many results from the literature can be treated from the view of ideals theory in a more simple way. Also, we offer a representation theorem for the spectrum of L. Finally, we study some types of ideals and we mention the possibility to use the theory of ideals in residuated lattices in order to develop applications in medical diagnosis.