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1 Varieties generated by t-norms

By Carol L. Walker

Abstract

Abstract — We consider generalized t-norms on distributive lattices and investigate properties of the variety of these algebras. We compare this with the variety generated by a strict t-norm. Two algebraic systems are isomorphic if there is a oneto-one mapping from one onto the other that preserves all of the operations. If the speci ed mathematical structure of an object is all that is used in an application, then isomorphic objects are interchangeable and the choice should not in uence the quality of the model. So for many applications, the main concern is determining which systems are isomorphic and which are not. In some fuzzy logic applications, however, the key properties depend on the equations satis ed by an algebraic system. Nonisomorphic algebras can satisfy exactly the same equations, and can, for example, lead to the same propositional logic. The class of all algebras of the same type that satisfy a particular set of equations is a variety. Let I = ([0; 1] ; ^; _; 0; 1), the unit interval with minimum and maximum determined by the usual order. This is an algebra of type (2; 2; 0; 0), meaning that it has two binary operations and two nullary operations (constants). Let x 0 = 1 x. The algebras (I; 0) an

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.353.2322
Provided by: CiteSeerX
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