On Decidability of T-norm-Based Equational Theories

The aim of this work is to show that the universal theory R of real closed fields [1] is interpretable in the equational theory of LPi1/2-algebras [2,3,6,7,8], and viceversa. Since R enjoys quantifier elimination, we will obtain that the full theory of R is interpretable in LPi1/2. This will also yield that any function definable in R is definable in LPi1/2. As a consequence of this construction we provide a description of (leftcontinuous) t-norms [5] definable in LPi1/2. In particular we can find a complete characterization of definable continuous t-norms. Theorem A continuous t-norm is definable iff it can be represented as a finite ordinal sum. This is due to the fact that since the set of idempotent elements of a tnorm is definable, by the properties of real closed fields it must be a Boolean combination of semialgebraic sets [1]. Negative results are also given for the definability of left continuous tnorms, i.e.: a definable left continuous t-norm cannot have a dense set or an infinite discrete set of discontinuities. However, many well-known leftcontinuous t-norms obtained by some construction methods [4] are definable in LPi1/2. Hence, we directly have the following theorem. Theorem The class of definable (left-continuous) t-norms is closed under (finite) ordinal sum, rotation, annihilation and rotation-annihilation. Now, let ∗ be a definable left-continuous t-norm, and →∗ its residuum. Then [0, 1]A∗ = ([0, 1], ∗,→∗,∧,∨) is a commutative, bounded, integral, residuated lattice. Suppose that the class of A∗-algebras is generated by [0, 1]A∗ . Lemma Let ∗ be a definable left-continuous t-norm. Then in [0, 1]LPi1/2 there is a definable structure isomorphic to [0, 1]A∗ . The left-continuous t-norm ∗ is definable in LPi1/2, and so is its residuum, hence they have as corresponding LPi1/2-functions ∗′ and →∗′ . Then, we can define a translation ◦ from [0, 1]A∗-terms into LPi1/2-terms so that • (x ∗ y)◦ is x ∗′ y • (x →∗ y)◦ is x →∗′ y.