Abstract
Power structures are obtained by lifting some mathematical structure (operations, relations, etc.) from an universe X to its power set \({\mathcal{P}(X)}\) . A similar construction provides fuzzy power structures: operations and fuzzy relations on X are extended to operations and fuzzy relations on the set \({\mathcal{F}(X)}\) of fuzzy subsets of X. In this paper we study how this construction preserves some properties of fuzzy sets and fuzzy relations (similarity, congruence, etc.). We define the notions of good, very good, Hoare good and Smith good fuzzy relation and establish some connections between them, generalizing some results of Brink, Bošnjak and Madarász on power structures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bělohlávek R.: Fuzzy Relational Systems. Foundations and Principles. Kluwer, Dordrecht (2002)
Bělohlávek R., Vychodil V.: Algebras with fuzzy equalities. Fuzzy Sets Syst. 157, 161–201 (2006)
Bošnjak I., Madarász R.: Power algebras and generalized quotient algebras. Algebra Universalis 45, 179–189 (2001)
Bošnjak I., Madarász R.: Good quotient algebras and power algebras. Novi Sad J. Math. 29, 71–84 (1999)
Bošnjak I., Madárasz R.: On some classes of good quotient relations. Novi Sad J. Math. 32, 131–140 (2002)
Brink C.: Power structures. Algebra Universalis 30, 177–216 (1993)
Brink C., Rewitzki I.: A Paradigm for Program Semantics. Power Strctures and Duality. CSLI Publications, Stanford (2001)
Burris S., Sankappanavar H.P.: A Course in Universal Algebra. Springer, New York (1981)
Dubois D., Prade H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)
Gautam N.D.: The validity of equations of complex algebras. Arch. Math. Logik Grundleg. 3, 117–124 (1957)
Grätzer G.: Universal algebra. Springer, Berlin (1979)
Hájek P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)
Klement E.P., Mesiar R., Pap E.: Triangular Norms. Kluwer, Dordrecht (2000)
Kuraoka T., Suzuki N.Y.: Lattice of fuzzy subalgebras in universal algebra. Algebra Universalis 47, 223–237 (2002)
Madarász R.: Generalized factor algebras. Matem. Bilten 18, 29–38 (1994)
Madarász R.: Remarks on power structures. Algebra Universalis 34, 179–184 (1995)
Schweizer B., Sklar A.: Statistical metric spaces. Pac. J. Math. 10, 312–334 (1960)
Zadeh L.A.: Similarity relations and fuzzy ordering. Inform. Sci. 3, 177–200 (1971)