Strictly join irreducible varieties of residuated lattices

TL;DR

This paper investigates the structure of join irreducible subvarieties within residuated lattices, providing characterizations and exploring their connections with well-connected algebras, especially in the context of representable and linear varieties.

Contribution

It introduces new characterizations of strictly join irreducible varieties in residuated lattices, including basic hoops and MTL-algebras, using generalized rotation and weakened completeness notions.

Findings

Characterization of strictly join irreducible varieties of basic hoops

Use of generalized rotation construction for MTL-algebras

Results on linear varieties of residuated lattices

Abstract

We study (strictly) join irreducible varieties in the lattice of subvarieties of residuated lattices. We explore the connections with well-connected algebras and suitable generalizations, focusing in particular on representable varieties. Moreover we find weakened notions of Hallden completeness that characterize join irreducibility. We characterize strictly join irreducible varieties of basic hoops, and use the generalized rotation construction to find strictly join irreducible varieties in subvarieties of MTL-algebras. We also obtain some general results about linear varieties of residuated lattices, with a particular focus on representable varieties, and a characterization for linear varieties of basic hoops.