FUZZY ORDERED SETS AND DUALITY FOR FINITE FUZZY DISTRIBUTIVE LATTICES

Document Type: Research Paper

Authors

1 Department of Mathematics, M’Sila University, P.O. Box 166, M’Sila 28000, Algeria

2 Department of Mathematics, Yazd University, Yazd, Iran

Abstract

The starting point of this paper is given by Priestley’s papers,
where a theory of representation of distributive lattices is presented. The purpose
of this paper is to develop a representation theory of fuzzy distributive
lattices in the finite case. In this way, some results of Priestley’s papers are
extended. In the main theorem, we show that the category of finite fuzzy
Priestley spaces is equivalent to the dual of the category of finite fuzzy distributive
lattices. Several examples are also presented.

Keywords


[1] D. Adnadjevic,Dimension of fuzzy ordered sets, Fuzzy Sets and Systems, 67 (1994), 349-357.

[2] G. Birkkhoff,Lattice theory, Colloquium Publications, Third Edition, Amer. Math. Soc., RI,1967.

[3] U. Bodenhofer,Representations and constructions of similarity-based fuzzy orderings, Fuzzy Sets and Systems,137(2003), 113-137.

[4] R. A. Borzooei and M. Bakhshi,T-Fuzzy congruences and T-fuzzy filters of a BI-algebra,Iranian Journal of Fuzzy Systems,6(4) (2009), 37-47.

[5] B. A. Davey and H. A. Priestley, Introduction to lattices and order, Second Edition, Cambridge University Press, Cambridge, 2002.

[6] J. C. Fodor and S. Ovchinnikov,On aggregation of T-transitivity fuzzy binary relations, Fuzzy Sets and Systems,72 (1995), 135-145.

[7] E. P. Klement and R. Mesiar,Logical, algebraic, analytic and probabilistic aspects of triangular norms, Elsevier, 2005.

[8] S. Kundu,Similarity relations, fuzzy linear orders and fuzzy partial orders, Fuzzy Sets and Systems,109 (2000), 419-428.

[9] H. Lai and D. Zhang,Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems, 157 (2006),1865-1885. 

[10] S. Lee, K. H. Lee and D. Lee,Order relation for type-2 fuzzy values, J. Tsinghua Sci. Technol.,8(2003), 30-36.

[11] H. A. Priestley,Representation of distributive lattices by means of ordered Stone spaces, Bull.London Math. Soc.,2 (1970), 186-190.

[12] H. A. Priestley,Ordered topological spaces and the representation of distributive lattices,Proc. London. Math. Soc.,24(3) (1972), 507-530.

[13] B. ˇSeˇselja,Lattice of partially ordered fuzzy subalgebras, Fuzzy Sets and Systems, 81 (1996),265-269. 

[14] B. ˇSeˇselja and A. Tepavˇcevi´c,Representing ordered structures by fuzzy sets: An overview,Fuzzy Sets and Systems,136 (2003), 21-39.

[15] M. H. Stone,Topological representations of distributive lattices and Brouwerian logics, ˇCasopis P˘est. Math.,67(1937), 1-25.

[16] P. Venugopalan,Fuzzy ordered sets, Fuzzy Sets and Systems, 46 (1992), 221-226.

[17] L. A. Zadeh,Similarity relation and fuzzy ordering, Information Sciences, 3 (1971), 177-200.