Bounded commutative residuated lattices with a retraction term

BUSANICHE, MANUELA

Denver

Congreso; BLAST; 2018

University of Denver

Substructural logics encompass many of the interesting nonclassical logics: intuitionistic logic, fuzzy logic, relevance logic, linear logic, besides including classical logic as a limit case. Residuated lattices are thealgebraic semantics of substructural logics, that is why their investigation is one of the main tools to understand and study those logical systems uniformly. But the multitude of different structures makes the study fairly complicated, thus the investigation of interesting subvarieties of residuated lattices is an appealing problem to address.In this talk we study subvarieties of bounded commutative residuated lattices with a retraction term. Many authors have already studied the case when the image of the retraction is a Boolean algebra. Now we investigate the structure of algebras with a retraction onto a hyperarchimedian MV-algebra, which includes the Boolean case. We introduce the notion of generalized rotation of a residuated lattice, and after showing that they have an MV-retraction, we characterize the varieties these generalized rotations generate. These varieties include among others:Product algebras, Stonean residuated lattices, BLN-algebras, perfect and bipartite MTL-algebras, Nilpotent Minimum algebras, Regular Nelson residuated lattices, SBP0-algebras.The purpose is to use the image and the kernel of the retraction term to represent each algebra by simpler and better-known structures: we present a categorical equivalence between our varieties generated by generalized rotations and categories whose objects are triples formed by two algebras and a connecting map. The categorical equivalence helps us to understand the structure of these novel algebras.Our construction has a dual aim: on one hand, it provides a common framework to compare classes of algebras treated independently in previous studies. On the other hand, it allows us to obtain particular results for important well-known classes of algebras and also new varieties that are certainly worth further investigation.