On some varieties of MTL-algebras

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Authors:
Carles Noguera
Francesc Esteva
Joan Gispert
Title: On some varieties of MTL-algebras
Journal: Logic Journal of the IGPL
Volume 13
Number 4
Pages: 443-466
Year: 2005
Preprint




Abstract

The study of perfect, local and bipartite IMTL-algebras presented in a previous paper is generalized in this paper to the general non-involutive case, i.e. to MTL-algebras. To this end we describe the radical of MTL-algebras and characterize perfect MTL-algebras as those for which the quotient by the radical is isomorphic to the two-element Boolean algebra, and a special class of bipartite MTL-algebras, 0, as those for which the quotient by the radical is a Boolean algebra. We prove that 0 is the variety generated by all perfect MTL-algebras and give some equational bases for it. We also introduce a new way to build MTL-algebras by adding a negation fixpoint to a perfect algebra and also by adding some set of points whose negation is the fixpoint. Finally, we consider the varieties generated by those algebras, giving equational bases for them, and we study which of them define a fuzzy logic with standard completeness theorem.