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On k-cyclic SHn-algebra

Fernandez A.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2011

Vol. 59, nr 3

303-304

In this work we consider a new class of algebra called k-cyclic SHn-algebra (A, T) where A is an SHn-algebra and T is a lattice endomorphism such that Tk(x) = x, for all x, k is a positive integer. The main goal of this paper is to show a Priestley duality theorem for k-cyclic SHn-algebra.

n x m Łukasiewicz algebras with parameters

Sanza C.

Reports on Mathematical Logic

2006

Nr 40

83-106

Matrix Łukasiewicz algebras were introduced by W. Suchoń in 1975 (Matrix Łukasiewicz Algebras, Reports on Mathematical Logic 4 (1975), 91-104). In this paper n x m{valued Łukasiewicz algebras with negation (or NSnxm algebras are defined and investigated. These algebras constitute an extension of those given by W. Suchoń and in m = 2 case they coincide with n valued Łukasiewicz algebras. Firstly, some of the main results established for matrix Łukasiewicz algebras are extended to NSnxm algebras. In particular, a functional representation theorem is given. Next, NSnxm congruences are determined by taking into account an implication operation which is defined on these algebras. In addition, it is proved that the class of NSnxm algebras is a variety. Besides, subdirectly irreducible algebras are characterized. As a consequence, it is shown that this variety is semisimple and locally finite. Finally, the algebra which generates the variety of NSnxm algebras is obtained and an equational base for the latter is determined.

Lipschitz continuity of fuzzy controller

Bartova D., Kukal J., Majerova D.

Archives of Control Sciences

2006

Vol. 16, no. 1

19-35

Any traditional fuzzy controller performs a sequence of three processes: fuzzification, control algorithm and defuzzification. It is useful when the contoller exhibits continuous behavior with constrained output and sensitivity. After the normalization of controller inputs and outputs into the interval [0;1], we designed the fuzzy controller to be Lipschitz continuous, which implies the constrained sensitivity of the controller. Łukasiewicz algebra enriched by ŁAsqrt was used for the realization of the proposed fuzzy controller. The realization of fuzzification and control algotithm is trivial. The only problem is in the defuzzification. Neither Mamdani nor Larsen approaches are continuous in general. Both MOM and COG technoques generate discontinuous output behaviour. That is why we developed a new defuzzification method based on Łukasiewicz algebra. Thus, the proposed technique of defuzzification is based on propositional logic and it helps to realize a class of Lipschitz continuous fuzzy controllers. the controllers were realized in the Matlab environment.