Homomorphisms between fuzzy information systems revisited

Recently, Wang et al. discussed the properties of fuzzy information systems under homomorphisms in the paper [C. Wang, D. Chen, L. Zhu, Homomorphisms between fuzzy information systems, Applied Mathematics Letters 22 (2009) 1045-1050], where homomorphisms are based upon the concepts of consistent functions and fuzzy relation mappings. In this paper, we classify consistent functions as predecessor-consistent and successor-consistent, and then proceed to present more properties of consistent functions. In addition, we improve some characterizations of fuzzy relation mappings provided by Wang et al.Comment: 10 page

Quantale Modules and their Operators, with Applications

The central topic of this work is the categories of modules over unital quantales. The main categorical properties are established and a special class of operators, called Q-module transforms, is defined. Such operators - that turn out to be precisely the homomorphisms between free objects in those categories - find concrete applications in two different branches of image processing, namely fuzzy image compression and mathematical morphology

Homomorphisms on the monoid of fuzzy implications and the iterative functional equation I(x,I(x,y))=I(x,y)

Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications, called the ⊛⊛-composition, from a given pair of fuzzy implications [Representations through a Monoid on the set of Fuzzy Implications, Fuzzy Sets and Systems, 247, 51-67]. However, as with any generation process, the ⊛⊛-composition does not always generate new fuzzy implications. In this work, we study the generative power of the ⊛⊛-composition. Towards this end, we study some specific functional equations all of which lead to the solutions of the iterative functional equation I(x,I(x,y))=I(x,y)I(x,I(x,y))=I(x,y) involving fuzzy implications which has been studied extensively for different families of fuzzy implications in this very journal, see [Information Sciences 177, 2954–2970 (2007); 180, 2487–2497 (2010); 186, 209–221 (2012)]. In this work, unlike in other existing works, we do not restrict the solutions to a particular family of fuzzy implications. Thus we take an algebraic approach towards solving these functional equations. Viewing the ⊛⊛-composition as a binary operation ⊛⊛ on the set II of all fuzzy implications one obtains a monoid structure (I,⊛)(I,⊛) on the set II. From the Cayley’s theorem for monoids, we know that any monoid is isomorphic to the set of all right translations. We determine the complete set KK of fuzzy implications w.r.t. which the right translations also become semigroup homomorphisms on the monoid (I,⊛I,⊛) and show that KK not only answers our questions regarding the generative power of the ⊛⊛-composition but also contains many as yet unknown solutions of the iterative functional equation I(x,I(x,y))=I(x,y)I(x,I(x,y))=I(x,y)

The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems

We prove that an $\omega$-categorical core structure primitively positively interprets all finite structures with parameters if and only if some stabilizer of its polymorphism clone has a homomorphism to the clone of projections, and that this happens if and only if its polymorphism clone does not contain operations $\alpha$, $\beta$, $s$ satisfying the identity $\alpha s(x,y,x,z,y,z) \approx \beta s(y,x,z,x,z,y)$. This establishes an algebraic criterion equivalent to the conjectured borderline between P and NP-complete CSPs over reducts of finitely bounded homogenous structures, and accomplishes one of the steps of a proposed strategy for reducing the infinite domain CSP dichotomy conjecture to the finite case. Our theorem is also of independent mathematical interest, characterizing a topological property of any $\omega$-categorical core structure (the existence of a continuous homomorphism of a stabilizer of its polymorphism clone to the projections) in purely algebraic terms (the failure of an identity as above).Comment: 15 page