Negation and affirmation: the role of involutive negators

Strict negators and automorphisms are prevalently used to fuzzify the Boolean negation and affirmation. Involutive negators are of particular interest. Every monotone [0,1] → [0,1] bijection is a composition of at most four involutive negators. Involutive negators are geometrically recognized by the symmetry of their graph w.r.t. the first bisector. If the graph of an automorphism has an alternating behavior, we can generate the automorphism by a pair of involutive negators.     

An Efficient Algebraic Algorithm for the Geometric Completion to Involution

 We describe an adaption of a differential algebraic completion algorithm for linear systems of partial differential equations that allows us to deduce intrinsic differential geometric information like the number of prolongations and projections needed for the completion. This new hybrid algorithm represents a much more efficient realisation of the classical Cartan–Kuranishi completion than previous purely geometric ones. A classical problem in geometric completion theory is the existence of δ-singular coordinate systems in which the algorithms do not terminate. We develop a new and a very simple criterion for δ-singularity based on a comparison of the Janet and the Pommaret division. This criterion can also be used for the direct construction of δ-regular coordinates. Received: July 28, 2000; revised version: October 16, 2001     

Determining singular solutions of polynomial systems via symbolic-numeric reduction to geometric involutive forms

We present a method based on symbolic-numeric reduction to geometric involutive form to compute the primary component of and a basis of Max Noether space for a polynomial system at an isolated singular solution. The singular solution can be known exactly or approximately. For the case where the singular solution is known with limited accuracy, we then propose a generalized quadratic Newton iteration for refining it to high accuracy.     

Noether normalization guided by monomial cone decompositions

This paper explains the relevance of partitioning the set of standard monomials into cones for constructing a Noether normalization for an ideal in a polynomial ring. Such a decomposition of the complement of the corresponding initial ideal in the set of all monomials–also known as a Stanley decomposition–is constructed in the context of Janet bases, in order to come up with sparse coordinate changes which achieve Noether normal position for the given ideal.     

Solving polynomial systems via symbolic-numeric reduction to geometric involutive form

We briefly survey several existing methods for solving polynomial systems with inexact coefficients, then introduce our new symbolic-numeric method which is based on the geometric (Jet) theory of partial differential equations. The method is stable and robust. Numerical experiments illustrate the performance of the new method.     

On fuzzy rough sets based on tolerance relations

Dubois and Prade (1990) [1] introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets, which was originally proposed by Pawlak (1982) [8]. Later, Radzikowska and Kerre introduced the so-called (I,T)-fuzzy rough sets, where I is an implication and T is a triangular norm. In the present paper, by using a pair of implications (I,J), we define the so-called (I,J)-fuzzy rough sets, which generalize the concept of fuzzy rough sets in the sense of Radzikowska and Kerre, and that of Mi and Zhang. Basic properties of (I,J)-fuzzy rough sets are investigated in detail.     

Algebraic structures for fuzzy numbers from categorial point of view

Using a classic algebraic construction additive and multiplicative structures (as commutative monoids) for fuzzy numbers are obtained. Moreover, we realize here an isomorphism between these structures.     

Complete involutive rewriting systems

Given a monoid string rewriting system M, one way of obtaining a complete rewriting system for M is to use the classical Knuth–Bendix critical pairs completion algorithm. It is well-known that this algorithm is equivalent to computing a noncommutative Gröbner basis for M. This article develops an alternative approach, using noncommutative involutive basis methods to obtain a complete involutive rewriting system for M.