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Abstract: The algebra of truth values for fuzzy sets of type-2 consists of all mappings from the unit interval into itself, with operations certain convolutions of these mappings with respect to pointwise max and min. This algebra generalizes the truth-value algebras of both type-1 and of interval-valued fuzzy sets, and has been studied rather extensively both from a theoretical and applied point of view. This paper addresses the situation when the unit interval is replaced by two finite chains. Most of the basic theory goes through, but there are several special circumstances of interest. These algebras are of interest on two counts, both as special cases of bases for fuzzy theories, and as mathematical entities per se. [Copyright &y& Elsevier]

The purpose of this paper is to give a straightforward mathematical treatment of algebras of fuzzy truth values for type-2 fuzzy sets. [Copyright &y& Elsevier]

In this paper we consider the rth powers of a continuous t-norm for positive real numbers r, generalizing the notion of the diagonal (2nd power). We identify the increasing functions from the unit interval to itself that are realized as the rth power of some continuous t-norm. The strict and nilpotent cases are described in detail. [Copyright &y& Elsevier]

We define a 2-category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions. We enrich this category with additional monoidal and involutive structure coming from t-norms and negations on the unit interval. We develop the basic properties of this category and consider its relation to other familiar categories. A discussion is made of extending these results to the setting of type-2 fuzzy sets. [ABSTRACT FROM AUTHOR]

Abstract: The lattice of upper semicontinuous convex normal functions with convolution ordering arises in studies of type-2 fuzzy sets. In 2002, Kawaguchi and Miyakoshi [Extended t-norms as logical connectives of fuzzy truth values, Multiple-Valued Logic 8(1) (2002) 53–69] showed that this lattice is a complete Heyting algebra. Later, Harding et al. [Lattices of convex, normal functions, Fuzzy Sets and Systems 159 (2008) 1061–1071] gave an improved description of this lattice and showed it was a continuous lattice in the sense of Gierz et al. [A Compendium of Continuous Lattices, Springer, Berlin, 1980]. In this note we show the lattice is isomorphic to the lattice of decreasing functions from the real unit interval to the interval under pointwise ordering, modulo equivalence almost everywhere. This allows development of further properties of . It is shown that is completely distributive, is a compact Hausdorff topological lattice whose topology is induced by a metric, and is self-dual via a period two antiautomorphism. We also show the lattice has another realization of natural interest in studies of type-2 fuzzy sets. It is isomorphic to a quotient of the lattice L of all convex normal functions under the convolution ordering. This quotient identifies two convex normal functions if they agree almost everywhere and their intervals of increase and decrease agree almost everywhere. [Copyright &y& Elsevier]

Abstract: This paper addresses some questions about the variety generated by the algebra of truth values of type-2 fuzzy sets. Its principal result is that this variety is generated by a finite algebra, and in particular is locally finite. This provides an algorithm for determining when an equation holds in this variety. It also sheds light on the question of determining an equational axiomatization of this variety, although this problem remains open. [Copyright &y& Elsevier]

Abstract: The algebra of truth values of type-2 fuzzy sets is the set of all functions from the unit interval into itself, with operations defined in terms of certain convolutions of these functions with respect to pointwise max and min. This algebra has been studied rather extensively, both from a theoretical and from a practical point of view. It has a number of interesting subalgebras, and this paper is about the subalgebra of all convex normal functions, and closely related ones. These particular algebras are De Morgan algebras, and our concern is principally with their completeness as lattices. A special feature of our treatment is a representation of these algebras as monotone functions with pointwise order, making the operations more intuitive. [Copyright &y& Elsevier]

The Leximin procedure, introduced by Dubois and Prade in 1996, provides a methodology for adjudicating ties in aggregated values when using the min t-norm. Here, with the introduction of the LexiT procedure, we generalize these ideas to provide a tool for adjudicating ties when aggregating data using any t-norm. We then provide the LexiS procedure for adjudicating ties when aggregating arguments using t-conorms. [Copyright &y& Elsevier]

In this paper, we examine and compare De Morgan-, Kleene-, and Boolean-disjunctive and conjunctive normal forms and consider their role in fuzzy settings. In particular, we show that there are normal forms and truth tables for classical fuzzy propositional logic and interval-valued fuzzy propositional logic that are completely analogous to those for Boolean propositional logic. Thus, determining logical equivalence of two expressions in fuzzy propositional logic is a finite problem, and similarly for the interval-valued case. Turksen''s work on interval-valued fuzzy sets is examined in light of these results. [Copyright &y& Elsevier]