Associatively tied implications. (English) Zbl 1042.03021

The paper deals with algebras proper for the development of fuzzy logic. The basic algebra is a partially ordered set with a top element 1 endowed by an implication triple \((A, K, H)\) where \(A\) is an implication (antitone in the left argument, isotone in the right and has 1 as a left identity element), \(K\) is a conjunction and \(H\) is a forcing implication so that the adjointness condition holds: \[ \beta \leq A(\alpha, \gamma) \text{ iff } K(\alpha, \beta)\leq \gamma \text{ iff }\alpha \leq H(\beta, \gamma). \] An implication operator \(A\) on a complete lattice \(L\) is associatively tied if there is a binary operation \(T\) on \(L\) that ties \(A\), i.e. the identity \(A(\alpha, A(\beta, \gamma))= A(T(\alpha, \beta), \gamma)\) holds for all \(\alpha, \beta, \gamma\in L\). The authors show that there exists a binary operation \(T_A\) that ties \(A\). Properties of this operation are studied. Characterizations for the validity of associative tiedness for an implication \(A\) are sought.


03B52 Fuzzy logic; logic of vagueness
03G25 Other algebras related to logic
Full Text: DOI


[1] Abdel-Hamid, A.A.; Morsi, N.N.; Abdel-Hamid, A.A.; Morsi, N.N., On the relationship of extended necessity measures to implication operators on the unit interval, Part I: inform. sci., Part II: J. fuzzy math., 4, 715-736, (1996) · Zbl 0874.03028
[2] B. De Baets, Model implicators and their characterization, in: N. Steels (Ed.), Proc. 1st ICSC Internat. Symposium on Fuzzy Logic, ICSC, Academic Press, 1995, pp. A42-A49.
[3] K. Demirli, An extended framework for operator selection in generalized modus ponens, in: Proc. Internat. Conference on Systems, Man and Cybernetics, 1995, 357-363.
[4] K. Demirli, B. De Baets, A general class of residual operators, in: Seventh IFSA World Conference, Vol. I, Academia, 1997, pp. 271-276.
[5] Demirli, K.; De Baets, B., Basic properties of implicators in a residual framework, Tatra mt. math. publ., 16, 1-16, (1999) · Zbl 0949.03025
[6] Dilworth, R.P., Abstract residuation over lattices, Bull. amer. math. soc., 44, 262-268, (1938) · Zbl 0018.34104
[7] Dilworth, R.P.; Ward, N., Residuated lattices, Trans. amer. math. soc., 45, 335-354, (1939) · Zbl 0021.10801
[8] Dubois, D.; Prade, H., Fuzzy logics and the generalized modus ponens revisited, Internat. J. cybernet. systems, 15, 293-331, (1984) · Zbl 0595.03016
[9] Dubois, D.; Prade, H., Fuzzy-set-theoretic differences and inclusions and their use in the analysis of fuzzy equations, Control cybernet., 13, 141-148, (1984)
[10] Dubois, D.; Prade, H.; Dubois, D.; Prade, H., Fuzzy sets in approximate reasoning, part II: logical approaches, Fuzzy sets and systems, Fuzzy sets and systems, 40, 203-244, (1991) · Zbl 0722.03018
[11] Dubois, D.; Prade, H., The three semantics of fuzzy sets, Fuzzy sets and systems, 90, 141-150, (1997) · Zbl 0919.04006
[12] Fodor, J.C., On fuzzy implication operators, Fuzzy sets and systems, 42, 293-300, (1991) · Zbl 0736.03006
[13] Fodor, J.C., A new look at fuzzy connectives, Fuzzy sets and systems, 57, 141-148, (1993) · Zbl 0795.04008
[14] Fodor, J.C., Contrapositive symmetry of fuzzy implications, Fuzzy sets and systems, 69, 141-156, (1995) · Zbl 0845.03007
[15] J.C. Fodor, M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Academic Publishers, Dordrecht. · Zbl 0827.90002
[16] Gaines, B.R., Foundations of fuzzy reasoning, Internat. J. man – machine stud., 8, 623-668, (1978) · Zbl 0342.68056
[17] Gierz, G., A compendium of continuous lattices, (1980), Springer Berlin · Zbl 0452.06001
[18] Goguen, J., The logic of inexact concepts, Synthese, 19, 325-373, (1969) · Zbl 0184.00903
[19] Gottwald, S., Many-valued logic and fuzzy set theory, (), 5-89 · Zbl 0968.03062
[20] Gratzer, G., General lattice theory, (1978), Birkhauser Basel · Zbl 0385.06015
[21] Morsi, N.N., Propositional calculus under adjointness, Fuzzy sets and systems, 132, 91-106, (2002) · Zbl 1029.03012
[22] N.N. Morsi, Categories of L-topological spaces with localization at points, Part 2: Coreflectiveness in the category of L-topological spaces, in preparation.
[23] N.N. Morsi, E.M. Roshdy, Implication triples, Fuzzy Sets and Systems, submitted for publication.
[24] N.N. Morsi, E.A.A. Mohamed, M.S. El-Zekey, Propositional calculus for adjointness lattices, Mathware and Soft Computing, accepted for publication.
[25] Restall, G., An introduction to substructural logics, (2000), Routledge London, New York · Zbl 1028.03018
[26] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), North-Holland Amsterdam · Zbl 0546.60010
[27] E. Trillas, L. Valverde, On some functionally expressible implications for fuzzy set theory, Proc. 3rd Internat. Seminar on Fuzzy Set Theory, Linz, Austria, 1981, pp. 173-190. · Zbl 0498.03015
[28] Yager, R.R., An approach to inference in approximate reasoning, Internat. J. man – machine stud., 13, 323-338, (1980)
[29] Zadeh, L.A., Outline of a new approach to the analysis of complex systems and decision processes, IEEE trans. systems man cybernet., 9, 28-44, (1973) · Zbl 0273.93002
[30] Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1, 3-28, (1978) · Zbl 0377.04002
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