Michał Baczyński | University of Silesia in Katowice (original) (raw)

Papers by Michał Baczyński

Contrapositive Symmetry of Distributive Fuzzy Implications

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2002

Recently, we have examined the solutions of the system of the functional equations I(x, T(y, z)) ... more Recently, we have examined the solutions of the system of the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, I(y, z)) = I(T(x, y), z), where T : [0, 1]2 → [0, 1] is a strict t-norm and I : [0, 1]2 → [0, 1] is a non-continuous fuzzy implication. In this paper we continue these investigations for contrapositive implications, i.e. functions which satisfy the functional equation I(x, y) = I(N(y), N(x)), with a strong negation N : [0, 1] → [0, 1]. We show also the bounds for two classes of fuzzy implications which are connected with our investigations.

Some new solutions of the distributivity law I(x, S(y, z)) = S(I(x, y), I(x, z)) among R-implications and triangular conorms

The different distributivity laws, and among them the following one, I(x, S(y, z)) = S(I(x, y), I... more The different distributivity laws, and among them the following one, I(x, S(y, z)) = S(I(x, y), I(x, z)), where I is an R-implication — or more generally a fuzzy implication — and S is a t-conorm, were thoroughly investigated in past years. Jayaram and Rao in 2004 solved this equation for R-implications generated from nilpotent t-norms and for continuous t-conorms. In 2009, Baczyński and Jayaram extended these results to the case of R-implications generated from strict t-norms and continuous Archimedean t-conorms. It this article we consider the above distributivity law in the case when I is an R-implication generated from strict t-norm and S is just a t-conorm or even more general operator. We find that there are new solutions that are t-conorms, but not continuous and Archimedean ones. We show that some previous results presented by Baczyński and Jayaram can be obtained as corollaries of our new results.

Some functional equations connected to the distributivity laws for fuzzy implications and triangular conorms

Recently in some considerations connected with the distributivity laws of fuzzy implications over... more Recently in some considerations connected with the distributivity laws of fuzzy implications over triangular norms and conorms, the following functional equation appeared f(min(x + y, a)) = min(f(x) + f(y), b), (1) where a; b are finite or infinite nonnegative constants (see [1]). In [2] we considered a generalized version of this equation in the case when both a and b are finite, namely the equation f(m1(x + y)) = m2(f(x) + f(y)), where m1, m2 are functions defined on some finite intervals of ℝ satisfying additional assumptions. In this article we enhance the results from [2], [3] and consider generalized versions of the equation (1) in the cases when a or b is infinite. We show that some well known solutions of several functional equations, that we presented earlier in [1], [4], can be obtained as corollaries of these new facts.

On the Distributivity Equation I (x, U 1(y, z)) = U 2( I (x, y), I (x, z)) for Decomposable Uninorms (in Interval-Valued Fuzzy Sets Theory) Generated from Conjunctive Representable Uninorms

Modeling Decisions for Artificial Intelligence, 2014

In this paper we continue investigations connected with distributivity of implication operations ... more In this paper we continue investigations connected with distributivity of implication operations over decomposable (t-representable) operations. Our main goal is to show the general method of solving the following distributivity equation \(\mathcal{I}(x,\mathcal{U}_1(y,z)) = \mathcal{U}_2(\mathcal{I}(x,y),\mathcal{I}(x,z))\), when \(\mathcal{U}_1\), \(\mathcal{U}_2\) are decomposable uninorms (in interval-valued fuzzy sets theory) generated from two conjunctive representable uninorms. As a byproduct result we show all solutions of some functional equation related to this case.

Characterizations for the migrativity of uninorms over N-ordinal sum implications

Computational and Applied Mathematics

Characterizations on migrativity of continuous triangular conorms with respect to N-ordinal sum implications

Information Sciences

IEEE Transactions on Fuzzy Systems, 2019

It is widely accepted that distributivity properties play a key role in fuzzy research, especiall... more It is widely accepted that distributivity properties play a key role in fuzzy research, especially in fuzzy control. Making use of the solution of the autodistributivity functional equations, we give a characterisation of all the four types of distributivity of fuzzy implication. The necessary and sufficient condition for all the four distributive equation is that the operator belongs to the pliant operator class. This theorem leads to a new implication called preference implication. It is well known that there is no implication in (continuous-valued) fuzzy logic that satisfies all the properties that are valid in (classical) twovalued logic. We show that preference implication fulfills: (i) the law of contraposition, (ii) the T-conditionality, (iii) the ordering property, (iv) the exchange principle, (v) the law of importation, (vi) the identity principle and (vii) the general hypothetical reasoning. Preference implication has a ν parameter i.e., the fixed point of the negation. This ν value serves as a threshold and with this value we can go back by projection to the two-valued logic case. At the end of the article we indicate that the preference implication is closely related to the preference relation used in multicriteria decision making. We point that if the preference implication is multiplicative transitive and reciprocal, then the pliant system is reduced to some particular generator function of Dombi.

Studies in Fuzziness and Soft Computing, 2008

The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

International Journal of Approximate Reasoning, 2017

Interval-valued fuzzy implications Admissible order Interval-valued generalized modus ponens Inte... more Interval-valued fuzzy implications Admissible order Interval-valued generalized modus ponens Interval-valued strong equality index In this work we introduce the definition of interval-valued fuzzy implication function with respect to any total order between intervals. We also present different construction methods for such functions. We show that the advantage of our definitions and constructions lays on that we can adapt to the interval-valued case any inequality in the fuzzy setting, as the one of the generalized modus ponens. We also introduce a strong equality measure between interval-valued fuzzy sets, in which we take the width of the considered intervals into account, and, finally, we discuss a construction method for this measure using implication functions with respect to total orders.

Studies in Fuzziness and Soft Computing, 2000

This paper describes algebraic properties of fuzzy implications family and provides formulas for ... more This paper describes algebraic properties of fuzzy implications family and provides formulas for new fuzzy implications with their classification. First, we describe the lattice of fuzzy implications and its sublattices. Next, we examine properties of contrapositive and selfconjugate fuzzy implications. Finite sublattices are depicted by Hasse diagrams.

Fuzzy Sets and Systems, Sep 1, 2017

Recently, two new families of fuzzy implication functions called probabilistic implications and p... more Recently, two new families of fuzzy implication functions called probabilistic implications and probabilistic S-implications were introduced by Grzegorzewski [6, 7, 9]. They are based on conditional copulas and make a bridge between probability theory and fuzzy logic. In this paper we generalize these two classes and propose a new kind of construction methods for fuzzy implications which are based on an a priori given fuzzy implication I and a semicopula B.

Information Processing and Management of Uncertainty in Knowledge-Based Systems, 2020

In our contribution we give some remarks and conclusions regarding reasoning schemas used in appr... more In our contribution we give some remarks and conclusions regarding reasoning schemas used in approximate reasoning. Based on created computer tool for image customization we give some advices regarding FITA. Also, we show some facts regarding Bandler-Kohout subproduct and we present results for several inference schemas.

Selected Properties of Generalized Hypothetical Syllogism Including the Case of R-implications

Communications in Computer and Information Science, 2018

In this paper we investigate the generalized hypothetical syllogism (GHS) in fuzzy logic, which c... more In this paper we investigate the generalized hypothetical syllogism (GHS) in fuzzy logic, which can be seen as the functional equation \(\sup _{z\in [0,1]} T(I(x,z), I(z,y))=I(x,y)\), where I is a fuzzy implication and T is a t-norm. Our contribution is inspired by the article [Fuzzy Sets Syst 323:117–137 (2017)], where the author considered (GHS) when T is the minimum t-norm. We show several general results and then we focus on R-implications. We characterize all t-norms which satisfy (GHS) with arbitrarily fixed R-implication generated from a left-continuous t-norm.

Continuous R-Implications

In this work we have solved an open problem re- lated to the continuity of R-implications. We hav... more In this work we have solved an open problem re- lated to the continuity of R-implications. We have fully character- ized the class of continuous R-implications obtained from any arbi- trary t-norm. Using this result, we also determine the exact inter- section between the continuous subsets of R-implications and (S,N)- implications.

In this chapter we discuss some open problems related to fuzzy implications, which have either be... more In this chapter we discuss some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. In fact, this chapter also contains the answer for one of the open problems, which is hitherto unpublished. The recently solved problems are so chosen to reflect the importance of the problem or the significance of the solution. Finally, some other problems that still remain unsolved are stated for quick reference.

On Fuzzy Sheffer Stroke Operation

Artificial Intelligence and Soft Computing, 2018

The generalization of the classical logical connectives to the fuzzy logic framework has been one... more The generalization of the classical logical connectives to the fuzzy logic framework has been one of the main research lines since the introduction of fuzzy logic. Although many classical logical connectives have been already generalized, the Sheffer stroke operation has received scant attention. This operator can be used by itself, without any other logical operator, to define a logical formal system in classical logic. Therefore, the goal of this article is to present some initial ideas on the fuzzy Sheffer stroke operation in fuzzy logic. A definition of this operation in the fuzzy logic framework is proposed. Then, a characterization theorem in terms of a fuzzy conjunction and a fuzzy negation is presented. Finally, we show when we can obtain other fuzzy connectives from fuzzy Sheffer stroke operation.

Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011), 2011

Recently, in [4], we have discussed the following distributive equation of implications I(x, T 1 ... more Recently, in [4], we have discussed the following distributive equation of implications I(x, T 1 (y, z)) = T 2 (I(x, y), I(x, z)) over t-representable t-norms, generated from strict t-norms, in interval-valued fuzzy sets theory. In this work we continue these investigations, but with the assumption that T 1 is generated from nilpotent t-norms, while T 2 is generated from strict t-norms. As a byproduct result we show all solutions for the following functional equation f (min(u 1 +v 1 , a), min(u 2 +v 2 , a)) = f (u 1 , u 2) + f (v 1 , v 2) related to this case.

On the Distributivity Equation mathcalI(x,mathcalU1(y,z))=mathcalU2(mathcalI(x,y),mathcalI(x,z))\mathcal{I}(x,\mathcal{U}_1(y,z)) = \mathcal{U}_2(\mathcal{I}(x,y),\mathcal{I}(x,z))mathcalI(x,mathcalU_1(y,z))=mathcalU_2(mathcalI(x,y),mathcalI(x,z)) for Decomposable Uninorms (in Interval-Valued Fuzzy Sets Theory) Generated from Conjunctive Representable Uninorms

Lecture Notes in Computer Science, 2014

In this paper we continue investigations connected with distributivity of implication operations ... more In this paper we continue investigations connected with distributivity of implication operations over decomposable (t-representable) operations. Our main goal is to show the general method of solving the following distributivity equation \(\mathcal{I}(x,\mathcal{U}_1(y,z)) = \mathcal{U}_2(\mathcal{I}(x,y),\mathcal{I}(x,z))\), when \(\mathcal{U}_1\), \(\mathcal{U}_2\) are decomposable uninorms (in interval-valued fuzzy sets theory) generated from two conjunctive representable uninorms. As a byproduct result we show all solutions of some functional equation related to this case.

Functional Equations Involving Fuzzy Implications and Their Applications in Approximate Reasoning

Advances in Intelligent Systems and Computing, 2013

Research on fuzzy implications, where the truth values belong to the unit interval [0,1], are car... more Research on fuzzy implications, where the truth values belong to the unit interval [0,1], are carried out from the beginning of fuzzy set theory and fuzzy logic. In recent years, investigations has been deepened,which resulted in publishing some surveys [6] and two research monographs [1, 3] entirely devoted to this class of fuzzy connectives.

Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011), 2011

In order to avoid combinatorial rule explosion in fuzzy reasoning, in this work we explore the di... more In order to avoid combinatorial rule explosion in fuzzy reasoning, in this work we explore the distributive equations of implications. In details, by means of the section of I, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication I(x, T 1 (y, z)) = T 2 (I(x, y), I(x, z)), when T 1 is a continuous but not Archimedean triangular norm, T 2 is a continuous Archimedean triangular norm and I is an unknown function. Our methods of proof can be applied to the three other functional equations related closely to the distributive equation of implication.

Contrapositive Symmetry of Distributive Fuzzy Implications

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2002

Recently, we have examined the solutions of the system of the functional equations I(x, T(y, z)) ... more Recently, we have examined the solutions of the system of the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, I(y, z)) = I(T(x, y), z), where T : [0, 1]2 → [0, 1] is a strict t-norm and I : [0, 1]2 → [0, 1] is a non-continuous fuzzy implication. In this paper we continue these investigations for contrapositive implications, i.e. functions which satisfy the functional equation I(x, y) = I(N(y), N(x)), with a strong negation N : [0, 1] → [0, 1]. We show also the bounds for two classes of fuzzy implications which are connected with our investigations.

Some new solutions of the distributivity law I(x, S(y, z)) = S(I(x, y), I(x, z)) among R-implications and triangular conorms

The different distributivity laws, and among them the following one, I(x, S(y, z)) = S(I(x, y), I... more The different distributivity laws, and among them the following one, I(x, S(y, z)) = S(I(x, y), I(x, z)), where I is an R-implication — or more generally a fuzzy implication — and S is a t-conorm, were thoroughly investigated in past years. Jayaram and Rao in 2004 solved this equation for R-implications generated from nilpotent t-norms and for continuous t-conorms. In 2009, Baczyński and Jayaram extended these results to the case of R-implications generated from strict t-norms and continuous Archimedean t-conorms. It this article we consider the above distributivity law in the case when I is an R-implication generated from strict t-norm and S is just a t-conorm or even more general operator. We find that there are new solutions that are t-conorms, but not continuous and Archimedean ones. We show that some previous results presented by Baczyński and Jayaram can be obtained as corollaries of our new results.

Some functional equations connected to the distributivity laws for fuzzy implications and triangular conorms

Recently in some considerations connected with the distributivity laws of fuzzy implications over... more Recently in some considerations connected with the distributivity laws of fuzzy implications over triangular norms and conorms, the following functional equation appeared f(min(x + y, a)) = min(f(x) + f(y), b), (1) where a; b are finite or infinite nonnegative constants (see [1]). In [2] we considered a generalized version of this equation in the case when both a and b are finite, namely the equation f(m1(x + y)) = m2(f(x) + f(y)), where m1, m2 are functions defined on some finite intervals of ℝ satisfying additional assumptions. In this article we enhance the results from [2], [3] and consider generalized versions of the equation (1) in the cases when a or b is infinite. We show that some well known solutions of several functional equations, that we presented earlier in [1], [4], can be obtained as corollaries of these new facts.

On the Distributivity Equation I (x, U 1(y, z)) = U 2( I (x, y), I (x, z)) for Decomposable Uninorms (in Interval-Valued Fuzzy Sets Theory) Generated from Conjunctive Representable Uninorms

Modeling Decisions for Artificial Intelligence, 2014

In this paper we continue investigations connected with distributivity of implication operations ... more In this paper we continue investigations connected with distributivity of implication operations over decomposable (t-representable) operations. Our main goal is to show the general method of solving the following distributivity equation \(\mathcal{I}(x,\mathcal{U}_1(y,z)) = \mathcal{U}_2(\mathcal{I}(x,y),\mathcal{I}(x,z))\), when \(\mathcal{U}_1\), \(\mathcal{U}_2\) are decomposable uninorms (in interval-valued fuzzy sets theory) generated from two conjunctive representable uninorms. As a byproduct result we show all solutions of some functional equation related to this case.

Characterizations for the migrativity of uninorms over N-ordinal sum implications

Computational and Applied Mathematics

Characterizations on migrativity of continuous triangular conorms with respect to N-ordinal sum implications

Information Sciences

IEEE Transactions on Fuzzy Systems, 2019

It is widely accepted that distributivity properties play a key role in fuzzy research, especiall... more It is widely accepted that distributivity properties play a key role in fuzzy research, especially in fuzzy control. Making use of the solution of the autodistributivity functional equations, we give a characterisation of all the four types of distributivity of fuzzy implication. The necessary and sufficient condition for all the four distributive equation is that the operator belongs to the pliant operator class. This theorem leads to a new implication called preference implication. It is well known that there is no implication in (continuous-valued) fuzzy logic that satisfies all the properties that are valid in (classical) twovalued logic. We show that preference implication fulfills: (i) the law of contraposition, (ii) the T-conditionality, (iii) the ordering property, (iv) the exchange principle, (v) the law of importation, (vi) the identity principle and (vii) the general hypothetical reasoning. Preference implication has a ν parameter i.e., the fixed point of the negation. This ν value serves as a threshold and with this value we can go back by projection to the two-valued logic case. At the end of the article we indicate that the preference implication is closely related to the preference relation used in multicriteria decision making. We point that if the preference implication is multiplicative transitive and reciprocal, then the pliant system is reduced to some particular generator function of Dombi.

Studies in Fuzziness and Soft Computing, 2008

The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

International Journal of Approximate Reasoning, 2017

Interval-valued fuzzy implications Admissible order Interval-valued generalized modus ponens Inte... more Interval-valued fuzzy implications Admissible order Interval-valued generalized modus ponens Interval-valued strong equality index In this work we introduce the definition of interval-valued fuzzy implication function with respect to any total order between intervals. We also present different construction methods for such functions. We show that the advantage of our definitions and constructions lays on that we can adapt to the interval-valued case any inequality in the fuzzy setting, as the one of the generalized modus ponens. We also introduce a strong equality measure between interval-valued fuzzy sets, in which we take the width of the considered intervals into account, and, finally, we discuss a construction method for this measure using implication functions with respect to total orders.

Studies in Fuzziness and Soft Computing, 2000

This paper describes algebraic properties of fuzzy implications family and provides formulas for ... more This paper describes algebraic properties of fuzzy implications family and provides formulas for new fuzzy implications with their classification. First, we describe the lattice of fuzzy implications and its sublattices. Next, we examine properties of contrapositive and selfconjugate fuzzy implications. Finite sublattices are depicted by Hasse diagrams.

Fuzzy Sets and Systems, Sep 1, 2017

Recently, two new families of fuzzy implication functions called probabilistic implications and p... more Recently, two new families of fuzzy implication functions called probabilistic implications and probabilistic S-implications were introduced by Grzegorzewski [6, 7, 9]. They are based on conditional copulas and make a bridge between probability theory and fuzzy logic. In this paper we generalize these two classes and propose a new kind of construction methods for fuzzy implications which are based on an a priori given fuzzy implication I and a semicopula B.

Information Processing and Management of Uncertainty in Knowledge-Based Systems, 2020

In our contribution we give some remarks and conclusions regarding reasoning schemas used in appr... more In our contribution we give some remarks and conclusions regarding reasoning schemas used in approximate reasoning. Based on created computer tool for image customization we give some advices regarding FITA. Also, we show some facts regarding Bandler-Kohout subproduct and we present results for several inference schemas.

Selected Properties of Generalized Hypothetical Syllogism Including the Case of R-implications

Communications in Computer and Information Science, 2018

In this paper we investigate the generalized hypothetical syllogism (GHS) in fuzzy logic, which c... more In this paper we investigate the generalized hypothetical syllogism (GHS) in fuzzy logic, which can be seen as the functional equation \(\sup _{z\in [0,1]} T(I(x,z), I(z,y))=I(x,y)\), where I is a fuzzy implication and T is a t-norm. Our contribution is inspired by the article [Fuzzy Sets Syst 323:117–137 (2017)], where the author considered (GHS) when T is the minimum t-norm. We show several general results and then we focus on R-implications. We characterize all t-norms which satisfy (GHS) with arbitrarily fixed R-implication generated from a left-continuous t-norm.

Continuous R-Implications

In this work we have solved an open problem re- lated to the continuity of R-implications. We hav... more In this work we have solved an open problem re- lated to the continuity of R-implications. We have fully character- ized the class of continuous R-implications obtained from any arbi- trary t-norm. Using this result, we also determine the exact inter- section between the continuous subsets of R-implications and (S,N)- implications.

In this chapter we discuss some open problems related to fuzzy implications, which have either be... more In this chapter we discuss some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. In fact, this chapter also contains the answer for one of the open problems, which is hitherto unpublished. The recently solved problems are so chosen to reflect the importance of the problem or the significance of the solution. Finally, some other problems that still remain unsolved are stated for quick reference.

On Fuzzy Sheffer Stroke Operation

Artificial Intelligence and Soft Computing, 2018

The generalization of the classical logical connectives to the fuzzy logic framework has been one... more The generalization of the classical logical connectives to the fuzzy logic framework has been one of the main research lines since the introduction of fuzzy logic. Although many classical logical connectives have been already generalized, the Sheffer stroke operation has received scant attention. This operator can be used by itself, without any other logical operator, to define a logical formal system in classical logic. Therefore, the goal of this article is to present some initial ideas on the fuzzy Sheffer stroke operation in fuzzy logic. A definition of this operation in the fuzzy logic framework is proposed. Then, a characterization theorem in terms of a fuzzy conjunction and a fuzzy negation is presented. Finally, we show when we can obtain other fuzzy connectives from fuzzy Sheffer stroke operation.

Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011), 2011

Recently, in [4], we have discussed the following distributive equation of implications I(x, T 1 ... more Recently, in [4], we have discussed the following distributive equation of implications I(x, T 1 (y, z)) = T 2 (I(x, y), I(x, z)) over t-representable t-norms, generated from strict t-norms, in interval-valued fuzzy sets theory. In this work we continue these investigations, but with the assumption that T 1 is generated from nilpotent t-norms, while T 2 is generated from strict t-norms. As a byproduct result we show all solutions for the following functional equation f (min(u 1 +v 1 , a), min(u 2 +v 2 , a)) = f (u 1 , u 2) + f (v 1 , v 2) related to this case.

On the Distributivity Equation mathcalI(x,mathcalU1(y,z))=mathcalU2(mathcalI(x,y),mathcalI(x,z))\mathcal{I}(x,\mathcal{U}_1(y,z)) = \mathcal{U}_2(\mathcal{I}(x,y),\mathcal{I}(x,z))mathcalI(x,mathcalU_1(y,z))=mathcalU_2(mathcalI(x,y),mathcalI(x,z)) for Decomposable Uninorms (in Interval-Valued Fuzzy Sets Theory) Generated from Conjunctive Representable Uninorms

Lecture Notes in Computer Science, 2014

In this paper we continue investigations connected with distributivity of implication operations ... more In this paper we continue investigations connected with distributivity of implication operations over decomposable (t-representable) operations. Our main goal is to show the general method of solving the following distributivity equation \(\mathcal{I}(x,\mathcal{U}_1(y,z)) = \mathcal{U}_2(\mathcal{I}(x,y),\mathcal{I}(x,z))\), when \(\mathcal{U}_1\), \(\mathcal{U}_2\) are decomposable uninorms (in interval-valued fuzzy sets theory) generated from two conjunctive representable uninorms. As a byproduct result we show all solutions of some functional equation related to this case.

Functional Equations Involving Fuzzy Implications and Their Applications in Approximate Reasoning

Advances in Intelligent Systems and Computing, 2013

Research on fuzzy implications, where the truth values belong to the unit interval [0,1], are car... more Research on fuzzy implications, where the truth values belong to the unit interval [0,1], are carried out from the beginning of fuzzy set theory and fuzzy logic. In recent years, investigations has been deepened,which resulted in publishing some surveys [6] and two research monographs [1, 3] entirely devoted to this class of fuzzy connectives.

Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011), 2011

In order to avoid combinatorial rule explosion in fuzzy reasoning, in this work we explore the di... more In order to avoid combinatorial rule explosion in fuzzy reasoning, in this work we explore the distributive equations of implications. In details, by means of the section of I, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication I(x, T 1 (y, z)) = T 2 (I(x, y), I(x, z)), when T 1 is a continuous but not Archimedean triangular norm, T 2 is a continuous Archimedean triangular norm and I is an unknown function. Our methods of proof can be applied to the three other functional equations related closely to the distributive equation of implication.