On lattice valued up-sets and down-sets

Isotone and anti-isotone mappings from a poset into a complete lattice are investigated as lattice-valued up-sets and down-sets, respectively. Cuts of these are shown to be analogue crisp sub-posets of the domain: up-set or semi-filters and down-sets or semi-ideals. The collection of all lattice-valued up-sets (down-sets) of a poset is a complete lattice under the order inherited from the lattice. Among other results, for a collection of crisp up-sets (down-sets) of a poset, necessary and sufficient conditions are given under which this collection consists of cuts of a lattice valued up-set (down-sets). A generalization in the sense of closed fuzzy sets with respect to fuzzy relations is also carried out.

[1]  D. Sinha,et al.  A general theory of fuzzy arithmetic , 1990 .

[2]  Branimir Seselja,et al.  Reprresentation of Lattices by Fuzzy Sets , 1994, Inf. Sci..

[3]  Branimir Seselja,et al.  A note on a natural equivalence relation on fuzzy power set , 2004, Fuzzy Sets Syst..

[4]  Ramón Fuentes-González,et al.  Down and up operators associated to fuzzy relations and t-norms: A definition of fuzzy semi-ideals , 2001, Fuzzy Sets Syst..

[5]  Bernard De Baets,et al.  Transitivity Bounds in Additive Fuzzy Preference Structures , 2007, IEEE Transactions on Fuzzy Systems.

[6]  Marsha Chechik,et al.  Efficient Multiple-Valued Model-Checking Using Lattice Representations , 2001, CONCUR.

[7]  Andreja Tepavcevic,et al.  General form of lattice-valued fuzzy sets under the cutworthy approach , 2007, Fuzzy Sets Syst..

[8]  Radim Bělohlávek,et al.  Fuzzy Relational Systems: Foundations and Principles , 2002 .

[9]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[10]  Heiko Vogler,et al.  A note on cut-worthiness of recognizable tree series , 2008, Fuzzy Sets Syst..

[11]  Branimir Seselja,et al.  Characterization of L-Fuzzy Semi-Filters and Semi-Ideals , 2009, IFSA/EUSFLAT Conf..

[12]  Lotfi A. Zadeh,et al.  Similarity relations and fuzzy orderings , 1971, Inf. Sci..

[13]  Bernard De Baets,et al.  Additive decomposition of fuzzy pre-orders , 2007, Fuzzy Sets Syst..

[14]  Branimir Seselja,et al.  L-fuzzy covering relation , 2007, Fuzzy Sets Syst..

[15]  Andreja Tepavcevic,et al.  L-fuzzy lattices: an introduction , 2001, Fuzzy Sets Syst..

[16]  Lawrence M. Brown,et al.  Fuzzy sets as texture spaces, I. Representation theorems , 2000, Fuzzy Sets Syst..

[17]  Branimir Seselja,et al.  Completion of ordered structures by cuts of fuzzy sets: an overview , 2003, Fuzzy Sets Syst..

[18]  Heiko Vogler,et al.  Cut sets as recognizable tree languages , 2006, Fuzzy Sets Syst..

[19]  Martine De Cock,et al.  Openings and closures of fuzzy preorderings: theoretical basics and applications to fuzzy rule-based systems , 2003, Int. J. Gen. Syst..

[20]  Ali Jaballah,et al.  Uniqueness results in the representation of families of sets by fuzzy sets , 2006, Fuzzy Sets Syst..

[21]  J. Goguen.  L-fuzzy sets , 1967 .

[22]  Branimir Seselja,et al.  Representing ordered structures by fuzzy sets: an overview , 2003, Fuzzy Sets Syst..

[23]  Mustafa Demirci,et al.  An introduction to vague complemented ordered sets , 2007, Inf. Sci..