Arch. Math. Logic (2001) 40: 39–67 c  Springer-Verlag 2001 The LΠ and LΠ 1 2 logics: two complete fuzzy systems joining L ´ ukasiewicz and Product Logics Francesc Esteva 1 , Llu´ ıs Godo 1 , Franco Montagna 2 1 Institut d’Investigaci´ o en Intellig` encia Artificial (IIIA), Consejo Superior de Investiga- ciones Cientificas (CSIC), Campus Universitat Aut` onoma de Barcelona, s/n, 08193 Bellaterra, Spain (e-mail: {esteva; godo}@iiia.csic.es) 2 Dipartimento di Matematica, Universit` a degli Studi di Siena, Via del Capitano 15, 53100 Siena, Italy (e-mail: montagna@unisi.it) Received: 28 December 1998 / revised version: 11 May 1999 Abstract. In this paper we provide a finite axiomatization (using two fini- tary rules only) for the propositional logic (called LΠ ) resulting from the combination of Lukasiewicz and Product Logics, together with the logic obtained by from LΠ by the adding of a constant symbol and of a defin- ing axiom for 1 2 , called LΠ 1 2 . We show that LΠ 1 2 contains all the most important propositional fuzzy logics: Lukasiewicz Logic, Product Logic, G¨ odel’s Fuzzy Logic, Takeuti and Titani’s Propositional Logic, Pavelka’s Rational Logic, Pavelka’s Rational Product Logic, the Lukasiewicz Logic with ∆, and the Product and G¨ odel’s Logics with ∆ and involution. Stan- dard completeness results are proved by means of investigating the algebras corresponding to LΠ and LΠ 1 2 . For these algebras, we prove a theorem of subdirect representation and we show that linearly ordered algebras can be represented as algebras on the unit interval of either a linearly ordered field, or of the ordered ring of integers, Z. 1. Introduction As shown in [H98], in the literature of fuzzy logic, three systems emerge: the Logic of Lukasiewicz (cf. e.g. [COM95] or [CM97]), G¨ odel’s Fuzzy Logic (cf. [Go33]), and the Product Logic (cf. [HGE96]). These logics correspond to the main three continuous t-norms over the unit real interval, namely the Lukasiewicz conjunction ⊗, defined by x ⊗ y = max(x + y - 1, 0), the operation ∧, defined by x ∧ y = min(x,y), and the product ⊙, defined by