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The study of perfect, local and bipartite IMTL-algebras presented in [29] 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, BP 0 , as those for which the quotient by the radical is a Boolean algebra. We prove that BP 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.

Key words Algebraic logic, fuzzy logic, left-continuous t-norm, mathematical fuzzy logic, monoidal triangular norm based logic, MTL-algebra, nilpotent minimum logic, non-classical logic, residuated lattice, substructural logic, variety, weak nilpotent minimum logic, WNM-algebra. MSC (2000) 03G25, 03B52In this paper we carry out an algebraic investigation of the weak nilpotent minimum logic (WNM) and its t-norm based axiomatic extensions. We consider the algebraic counterpart of WNM, the variety of WNM-algebras (WNM) and prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM-chains, in particular varieties generated by standard WNM-chains, or equivalently t-norm based axiomatic extensions of WNM, and we study their standard completeness properties. We also characterize the generic WNM-chains, i. e. those that generate the variety WNM, and we give finite axiomatizations for some t-norm based extensions of WNM.

Chang's MV-algebras, on the one hand, are the algebras of the infinite-valued Lukasiewicz calculus and, on the other hand, are categorically equivalent to abelian latticeordered groups with a distinguished strong unit, for short, unital -groups. The latter are a modern mathematization of the time-honored euclidean magnitudes with an archimedean unit. While for magnitudes the unit is no less important than the zero element, its archimedean property is not even definable in first-order logic. This gives added interest to the equivalent representation of unital -groups via the equational class of MV-algebras. In this paper we survey several applications of this equivalence, and various properties of the variety of MV-algebras. PrologueIn the last twenty years the number of papers devoted to Chang's MV-algebras [14] has been increasing so rapidly that, since the year 2000 the AMS Classification Index contains the special item 06D35 for MV-algebras. To quote just a handful of books, the monograph [25] by Cignoli et al., is entirely devoted to MValgebras, Hajek's monograph [56] and Gottwald's book [54] devote ample space to these algebras. As shown in [37] and [94], MV-algebras also provide an interesting example of "quantum structures". The Handbook of Measure Theory [89] includes several chapters on MV-algebraic measure theory. As the Lindenbaum algebras of Lukasiewicz infinite-valued logic, MV-algebras are also considered in many surveys, e.g., [21,68,87]. In this paper we will present several main topics in the theory of MV-algebras which are of greater potential interest for the universal algebraist. We apologize to authors and readers for all omissions, mainly due to lack of space.

IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic interpretation to this construction, we generalize the concepts of perfect, bipartite and local algebra used in the classification of MV-algebras to the wider variety of IMTL-algebras and we prove that perfect algebras are exactly those algebras obtained from a prelinear semihoop by Jenei's disconnected rotation. We also prove that the variety generated by all perfect IMTL-algebras is the variety of the IMTL-algebras that are bipartite by every maximal filter and we give equational axiomatizations for it.