Fuzzy Sets and Systems 159 (2008) 1131 – 1152 www.elsevier.com/locate/fss Normal forms and free algebras for some extensions of MTL Stefano Aguzzoli a , ∗ , Brunella Gerla b a Dept. Computer Science, Università degli Studi di Milano, Via Comelico 39-41, 20135 Milano, Italy b DICOM, Università dell’Insubria, Via Mazzini 5, 21100 Varese, Italy Available online 8 December 2007 Abstract We introduce a semantical definition of minterms and maxterms which generalizes the usual notion in Boolean logic to a class of many-valued logics. We apply this notion to get normal forms for logics G, NM, NMG. Then we obtain a combinatorial description of the n-generated free algebras in the varieties constituting the algebraic semantics of those logics. Specifically, we represent via combinatorial posets the embedding of the n-generated free algebra into the direct product of all n-generated chains in the variety. © 2008 Elsevier B.V. All rights reserved. Keywords: Nilpotent minimum and Gödel logic; MTL; WNM; Algebraic semantics; Normal forms; Free algebras 1. Introduction In classical propositional logic, each formula over n variables is associated with a truth table, i.e. with a function from {0, 1} n to {0, 1}. The free Boolean algebra over n generators is the algebra of formulas with n variables, up to logical equivalence, and the functional completeness result for Boolean propositional logic states that the free Boolean algebra over n generators is the direct product of 2 n copies of the Boolean algebra {0, 1}. A standard way to obtain a formula associated with a given function f :{0, 1} n →{0, 1} is to write it in either disjunctive or conjunctive normal form. Disjunctive normal forms are nothing else than sums (∨) of minterms, where a minterm is a formula having a truth table that takes value 1 over exactly one point in {0, 1} n . In classical propositional logic, minterms are products (∧) of variables or negation of variables. Dually, conjunctive normal forms are product of maxterms, each maxterm being the sum of variables or negation of variables. If we enlarge the set of truth values, the situation changes. In particular, if the set of truth values is the interval [0, 1] there is no hope to establish any functional completeness result if we consider a propositional logic built on a finite or denumerable alphabet. Even in the case the free algebras are finite, they are not in general direct product of chains. The problem of characterizing the set of truth functions of a given logic is hence interesting. This amounts to finding concrete representations of free algebras in the variety that constitutes the algebraic semantics of the logic. The logics we shall deal with in this paper are schematic extensions of monoidal t-norm based logic (MTL, [11]) that is the logic of all left-continuous t-norms and their residua. In particular we shall focus on some logics in the hierarchy of extensions of weak nilpotent minimum logic (WNM, [11]) which in turn is a schematic extension of MTL whose corresponding algebraic variety is locally finite. ∗ Corresponding author. E-mail addresses: aguzzoli@dsi.unimi.it (S. Aguzzoli), brunella.gerla@uninsubria.it (B. Gerla). 0165-0114/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2007.12.003