Residuated Lattices Research Papers - Academia.edu (original) (raw)
We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective inspired by Hiroakira Ono's algebraic approach to substructural logics. We propose generalized BI algebras (GBI-algebras) as a common... more
We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective inspired by Hiroakira Ono's algebraic approach to substructural logics. We propose generalized BI algebras (GBI-algebras) as a common framework for algebras arising via " declarative resource reading " , intuitionistic generalizations of relation algebras and arrow logics and the distributive Lambek calculus with intuitionistic implication. Apart from existing models of BI (in particular, heap models and effect algebras), we also cover models arising from weakening relations, formal languages or more fine-grained treatment of labelled trees and semistructured data. After briefly discussing the lattice of subvarieties of GBI, we present a suitable duality for GBI along the lines of Esakia and Priestley and an algebraic proof of cut elimination in the setting of residuated frames of Galatos and Jipsen. We also show how the algebraic approach allows generic results on decidability, both positive and negative ones. In the final part of the paper, we gently introduce the substructural audience to some theory behind state-of-art tools, culminating with an algebraic and proof-theoretic presentation of (bi-) abduction.
Ideals in MV algebras are, by definition, kernels of homomorphism. An ideal is the dual of a filter in some special logical algebras but not in non-regular residuated lattices. Ideals in residuated lattices are defined as natural... more
Ideals in MV algebras are, by definition, kernels of homomorphism. An ideal is the dual of a filter in some special logical algebras but not in non-regular residuated lattices. Ideals in residuated lattices are defined as natural generalizations of ideals in MV algebras. Spec(L), the spectrum of a residuated lattice L, is the set of all prime ideals of L and it can be endowed with the spectral topology. The main scope of this paper is to characterize Spec(L), called the stable topology. In this paper, we introduce and investigate the notion of pure ideal in residuated lattices and using these ideals we study the related spectral topologies. Also, using the model of MV algebras, for a De Morgan residuated lattice L, we construct the Belluce lattice associated with L. This will provide informations about the pure ideals and the prime ideals space of L. So, in this paper we generalize some results relative to MV algebras to the case of residuated lattices.