Mayuka F. Kawaguchi

Composite fuzzy relational equations with non-commutative conjunctions

Abstract Through this study, the authors treat two kinds of fuzzy logical operators, that is, non-commutative conjunctions and their residual implications which include almost all of already-known functions in the field of fuzzy logic. Corresponding to such operators, two kinds of generalized compositions of fuzzy relations are introduced, and the solutions of new composite relational equations are given. This work reveals that the solutions include the results of the past researches in the field of fuzzy relational equations as special cases. Some specific numerical examples of the solutions of a given equation illustrate the mechanism of these compositions.

A calculation method for solving fuzzy arithmetic equations with triangular norms

The authors apply a previously developed calculation method using a digital representation to obtain an approximate solution to a fuzzy arithmetic equation. A t-norm and a phi -operator which is defined in connection with a given t-norm are summarized. Nonstandard operations based on the inf- phi convolution are discussed as a solution for the fuzzy arithmetic equation. Formulas involving the solution of the equations as well as the classifications of t-norms and phi -operators that are necessary for applying the formulas are presented. Some numerical examples are included.<<ETX>>

Generalized Extended t-Norms as t-Norms of Type 2

This research work focuses on the logical connectives for type 2 fuzzy logics. Especially, the operators which are obtained by extending continuous t-(co)norms to the case of fuzzy truth values by mean of the generalized extension principle are considered. The authors show that these operators named generalized extended t-(co)norms satisfy the definitions of t-(co)norms of type 2.

A necessary condition for solvability of fuzzy arithmetic equations with triangular norms

This paper treats a fuzzy arithmetic equation based on sup-(t-norm) convolution. The authors have investigated the properties of the procedure to solve the equation (i.e. inf- /spl phi/ convolution) and discussed its solvability. A necessary condition for solving the equations involving addition and multiplication has been derived as the main result of this work.<<ETX>>

Parameter formulae for fundamental operations of weakly non-interactive fuzzy numbers

D. Dubois and H. Prade (1981) introduced the concept of weakly noninteractive fuzzy numbers whose operations are based on the extension principle corresponding to each t-norm in place of the minimum operator. Some properties of weakly noninteractive fuzzy numbers and their practical method of calculation are investigated. Three parameters indicating the mean value and the left/right spreads of the fuzzy number are considered. Various parameter formulas for arithmetic operations and power function operation of certain kinds of weakly noninteractive fuzzy numbers involving no-interactive fuzzy numbers are presented. The formulas are applicable to both cases of the L-R fuzzy number of Dubois and Prade (1978) and an improved version of the calculation method using the digital representation. An attempt is made to classify general t-norms into the three classes from the viewpoint of the parameter formulas. The results of numerical experiments are shown for the formulas and the calculation method using the digital representation.<<ETX>>

Some algebraic properties of weakly non-interactive fuzzy numbers

Abstract This study focuses on the concept of sup-(t-norm) convulution, a generalized version of Zadehs extension principle. The authors treat the algebraic structure of fuzzy arithmetic based on the sup-(t-norm) convolution, and especially show some new properties comparing them with those of the conventional fuzzy arithmetic based on sup-min convolution.

Partially ordered set with residuated t-norm

We consider properties of partially ordered sets with residuated t-norm and show that 1. If (X;T,0,1) is a bounded partially ordered set with residuated t-norm T, then (X;*,0/sub X/,1/sub X/) is a bounded BCK-algebra with condition (S); 2. Conversely, if (B;*,0/sub B/,1/sub B/) is a bounded BCK-algebra with (S), then (B;T,0,1) is the bounded partially ordered set with residuated t-norm. This means that the class of all bounded partially ordered sets with residuated t-norm coincides with the class of all bounded BCK-algebras with condition (S). Since the class of these algebras forms a variety, the class of partially ordered sets with residuated t-norm is represented by only equations.

Some Properties of Generalized State Operators on Residuated Lattices

We define a generalized state operator σ on a residuated lattice X and a g-state residuated lattice (X,σ), and consider properties of g-state residuated lattices. We show that a characterization theorem of σ-filters and that the class F_σ (X) of all σ-filters of a g-state residuated lattice (X, σ) is a Heyting algebra. Moreover we prove that every g-state residuated lattice (X, σ) is isomprphic to a subdirect product of g-state residuated lattices {(X/P, σ/P)}_P∈Specσ(X), where Spec_σ(X) is the set of all prime σ-filters of (X, σ).

Construction of Associative Functions for Several Fuzzy Logics via the Ordinal Sum Theorem

In this report, the ordinal sum theorem of semigroups is applied to construct logical operations for several fuzzy logics. The generalized form of ordinal sum for fuzzy logics on [0, 1] is defined in order to uniformly express several families of logical operations. Then, the conditions in ordinal sums for various properties of logical operations are presented: for examples, the monotonicity, the location of the unit element, the left/right-continuity, or and/or-likeness. Finally, some examples to construct pseudo-uninorms by the proposed method are illustrated.

A correspondence between implicational fragment logics and fuzzy logics

This research report treats a correspondence between implicational fragment logics and fuzzy logics from the viewpoint of their algebraic semantics. The authors introduce monotone BI-algebras by loosening the axiomatic system of BCK-algebras. Also, we extend the algebras of fuzzy logics with weakly-associative conjunction from the case of the unit real interval to the case of a partially-ordered set. As the main result of this report, it is proved that the class of monotone BI-algebras with condition (S) coincides with the class of weakly-associative conjunctive algebras.