Abstract
The features of fuzzy lattices valued by lattices can be observed in the light of more general results from fuzzy algebraic and fuzzy relational structures. By approaching directly the problem of defining the notion of a fuzzy lattice, several directions can be taken. In this paper we present an idea and two of its variations to define what is referred to as L M fuzzy lattices. The idea is to fuzzify the membership functions of elements of the carrier of an ordinary, crisp, lattice. L M1 fuzzy lattices require for the cuts of the structure to be sublattices of the lattice whose carrier’s membership function has been the subject of fuzzification. More generally, L M2 fuzzy lattices require that the cuts are lattices themselves, not insisting on being substructures of the crisp lattice. The structure of the famines of cuts in both cases are presented, as well as an algorithm to construct L M fuzzy lattice with a given set of cuts. Alternative approaches to defining fuzzy lattices are discussed at the end of the paper.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ajmal, N., Thomas, K.V.: Fuzzy Lattices. Information Sciences, 79 (1994), 271–291.
Ajmal, N., Thomas, K.V.: The Lattices of Fuzzy Ideals of a Ring. Fuzzy Sets and Systems, 74 (1995), 371–379.
Birkhoff, G.: Lattice Theory, Amer. Math. Soc, Providence R.I., 1984.
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, Cambridge University Press, 1990.
Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980.
Goguen, J.A.: L-Fuzzy Sets. J. Math. Appli. 18 (1967), 145–174.
Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic, Prentice Hall, Upper Saddle River, NJ, 1995.
Rosenfield, A.: Fuzzy Groups. J. Math. Anal. Appl., 35 (1971), 512–517.
Šešelja, B., Tepavčević, A.: Partially Ordered and Relational Valued Algebras and Congruences. Review of Research, Fac. Sci., Univ. Novi Sad, Math. Ser. 23 (1993), 273–287.
Šešelja, B., Tepavčević, A.: Partially Ordered and Relational Valued Fuzzy Relations I. Fuzzy Sets and Systems, 72 (1995), 205–213.
Šešelja, B., Tepavčević, A.: Fuzzy Boolean Algebras. Automated Reasoning, IFIP Trans. A-19 (1992), 83–88.
Trajkovski, G.: An Approach Toward Defining L fuzzy lattices, Proc. NAFIPS’99, Pensacola Beach, FL, USA (1998), 221–225.
Trajkovski G., Fuzzy Relations and Fuzzy Lattices, M.Sc. Thesis, University “St. Cyril and Methodius” — Skopje, 1997.
Yager, R.R., Ovchinikov, S., Tong, R.M., Nguyen, H.T. (eds.): Fuzzy Sets and Applications: Selected papers by L.A. Zadeh, John Willey fc Sons, New York, 1987.
Yuan, B., Wu, W.: Fuzzy ideals on a distributive lattice. Fuzzy Sets and Systems, 35 (1990), 231–240.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Trajkovski, G., Čukić, B. (1999). On Two Types of L M Fuzzy Lattices. In: Reusch, B. (eds) Computational Intelligence. Fuzzy Days 1999. Lecture Notes in Computer Science, vol 1625. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48774-3_32
Download citation
DOI: https://doi.org/10.1007/3-540-48774-3_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66050-7
Online ISBN: 978-3-540-48774-6
eBook Packages: Springer Book Archive