On linear varieties of MTL-algebras

Abstract

In this paper, we focus on those varieties of MTL-algebras whose lattice of subvarieties is totally ordered. Such varieties are called linear. We show that a variety $${{\mathbb {L}}}$$ of MTL-algebras is linear if and only if each of its subvarieties is generated by one chain. We also study the order type of their lattices of subvarieties, and the structure of their generic chains. If $${\mathbb {L}}$$ is a linear variety with the finite model property, we have that the class of chains in $${\mathbb {L}}$$ is formed by either bipartite or simple chains. As a further result, we provide a complete classification of the linear varieties of BL-algebras. The more general case of MTL-algebras is out of reach, but nevertheless we classify all the linear varieties of WNM-algebras.