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The truth value algebra of type-2 fuzzy sets. Order convolutions of functions on the unit interval. (English) Zbl 1360.03003

Monographs and Research Notes in Mathematics. Boca Raton, FL: CRC Press (ISBN 978-1-4987-3527-8/hbk; 978-1-4987-3529-2/ebook). xix, 234 p. (2016).
The type-2 fuzzy sets are generalizations of classical fuzzy sets and interval-valued fuzzy sets. This book has 10 chapters, each chapter has its own exercises. In Chapter 1, the authors define the related truth value algebra \(M\) whose elements are selfmaps of the real unit interval with the basic operations given as convolutions, afterwards simplified in an opportune manner. The properties of these simplified operations are established in Chapter 2. Here, it is proved that \(M\) is a De Morgan-Birkhoff system and a bisemilattice with two partial orderings. Chapter 3 contains two important theorems which prove that type-1 fuzzy sets and interval-valued fuzzy sets are subalgebras of \(M\). Various subalgebras of \(M\) are also investigated, mainly the subalgebra \(L\) of the normal convex functions (here, normal function is understood as having supremum 1 and convex function is understood as either upper or lower function in the sense that a function \(f\) of \(M\) is lower (resp. upper) if \(x>1/2\) (resp., \(x<1/2\)) implies \(f(x)=0)\)). Chapter 4 is devoted to the study of the group \(\mathrm{Aut}(M)\) of the automorphisms of \(M\): the main theorem states that \(\mathrm{Aut}(M)\) is isomorphic to \(\mathrm{Aut}(I)\), where \(I\) is the truth value algebra of fuzzy sets. Further results concern characteristic subalgebras of \(M\). Among others, \(L\) is proved to be characteristic (i.e., a subalgebra \(A\) of \(M\) is characteristic if every automorphism of \(M\) restricts to give an automorphism of \(A\)), for further details, see, for instance, the works of the second and third author [Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 14, No. 6, 711–732 (2006; Zbl 1114.03053); ibid. 16, No. 5, 627–643 (2008; Zbl 1152.03331)]. In Chapter 5, the authors define the convolutions associated to a triangular norm and to a triangular conorm between two elements of \(M\) and prove several algebraic properties of these operations. The main result concerns the algebra \(L\): indeed if the triangular norm and the conorm are continuous, then the related associated convolutions are lattice-ordered over \(L\) (i.e., if they are increasing in the coordinates and distributive with respect to the internal operations of the bisemilattice of \(L\)). For further details, see, for instance, the work of the second and third author [Fuzzy Sets Syst. 149, No. 2, 309–347 (2005; Zbl 1064.03020)]. The results of Chapter 6 are extracted from the works of the authors [Fuzzy Sets Syst. 161, No. 9, 1343–1349 (2010; Zbl 1193.03077); ibid. 159, No. 9, 1061–1071 (2008; Zbl 1176.03033)]. Chapter 7 contains results extracted from the work of the same authors [ibid. 161, No. 5, 735–749 (2010; Zbl 1192.03051)]. In this chapter, it is proved that the variety generated by \(M\) coincides with the variety generated by its subalgebra \(E\) constituted of characteristic functions of sets. Furthermore, the authors prove that the same variety is generated by a finite De Morgan bisemilattice algebra with twelve elements. This allows to establish when an equation holds in \(M\) if it holds in this finite algebra. Further results involve \(M\) and the varieties generated by complex algebras of chains. The study of the variety generated by \(M\), considering only the two operations of the semilattice, continues in Chapter 8 where the authors prove that it coincides with the variety generated by a bichain \(B\) with 4 elements. This variety is widely studied in the context of Birkhoff systems and some conjectures are presented. Chapter 9 contains results on categories of fuzzy relations. Firstly, using results of Chapter 7, the authors show that the convolution associated to a continuous triangular norm assigned on matrices with entries from the subalgebra \(L\) is the multiplication of matrices which is associative and has an identity. This allows to define the category of type-2 fuzzy relations. Secondly, it is proved that such category is symmetric monoidal. As an interesting application, an example involving fuzzy controllers working with type-1 fuzzy matrices is extended to the type-2 setting in the context of this categorical approach. Chapter 10 deals the finite case, i.e. the algebra \(M\) is replaced by an algebra of functions sending a finite chain of \(n\) elements into a finite chain of \(m\) elements. Here, the authors study its subalgebras, its automorphisms, its convex normal functions and further properties are proved. A bibliography of 122 titles completes this book.

MSC:

03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03E72 Theory of fuzzy sets, etc.
03G25 Other algebras related to logic
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