Fuzzy order congruences on fuzzy posets

Document Type: Research Paper

Authors

1 College of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450000, China

2 College of Mathematics and Econometrics, Hunan University, Chang- sha, 410082, China

3 College of Information Science and Engineering, Hunan University, Changsha, 410082, China

Abstract

Fuzzy order congruences play an important role in studying the categorical
properties of fuzzy posets. In this paper, the correspondence between the fuzzy
order congruences and the fuzzy order-preserving maps is discussed. We focus on
the characterization of fuzzy order congruences on the fuzzy poset in terms of
the fuzzy preorders containing the fuzzy partial order. At last, fuzzy complete
congruences on fuzzy complete lattices are discussed.

Keywords


bibitem{ahs90} J. Adamek, H. Herrlich and G. Strecker, {it Abstract and
concrete categories: the joy of cats}, John Wiley and Sons, New York, 1990.

bibitem{rb02} R. Bv{e}lohl'avek, {it Fuzzy relational systems: foundations
and principles}, Kluwer Academic Publishers, Norwell, MA, USA, 2002.

bibitem{rb04} R. Bv{e}lohl'{a}vek, {it Concept lattices and order in fuzzy logic},
Ann. Pure Appl. Logic, {bf 128} (2004), 277--298.

bibitem{bt03} K. Blount and C. Tsinakis, {it The structure of residuated lattices},
Int. J. Algebr. Comput., {bf 13}textbf{(4)} (2003), 437--461.

bibitem{ub04} U. Bodenhofer, {it Applications of fuzzy orderings: an overview},
In: K. T. Atanasov, O. Hryniewicz, J. Kacprycz, eds., Soft Computing. Foundations
and Theoretical Aspects, EXIT, Warsaw, (2004), 81--95.

bibitem{ub07} U. Bodenhofer, B. De Baets and J. Fodor, {it A compendium of fuzzy
weak orders}, Fuzzy Sets and Systems, {bf 158} (2007), 811--829.

bibitem{csn98} I. Chajda and V. Sn'{a}v{s}el, {it Congruences in ordered sets},
Math. Bohem., {bf 123}textbf{(1)} (1998), 95--100.

bibitem{cib07} M. '{C}iri'{c}, J. Ignjatovi'{c} and S. Bogdanovi'{c},
{it Fuzzy equivalence relations and their equivalence classes}, Fuzzy Sets and Systems,
{bf 158} (2007), 1295--1313.

bibitem{dp02} B. A. Davey and H. A. Priestley, {it Introduction to lattices and
order}, Cambridge University Press, Cambridge, 2002.

bibitem{debm98} B. De Baets and R. Mesiar, {it T-partitions}, Fuzzy Sets and Systems,
{bf 97} (1998), 211--223.

bibitem{debm02} B. De Baets and R. Mesiar, {it Metrics and T-equalities},
J. Math. Anal. Appl., {bf 267} (2002) 531--547.

bibitem{mde03I} M. Demirci, {it Indistinguishability operators in measurement
theory, Part I: Conversions of indistinguishability operators with respect to
scales}, Internat. J. General Systems, {bf 32} (2003), 415--430.

bibitem{mde03II} M. Demirci, {it Indistinguishability operators in measurement
theory, Part II: Construction of indistinguishability operators based on
probability distributions}, Internat. J. General Systems, {bf 32} (2003), 431--458.

bibitem{mde04} M. Demirci and J. Recasens, {it Fuzzy groups, fuzzy functions and
fuzzy equivalence relations}, Fuzzy Sets and Systems, {bf 144} (2004), 441--458.

bibitem{lf01} L. Fan, {it A new approach to quantitative domain theory},
Electronic Notes in Theoretical Computer Science, {bf 45} (2001), 77--87.


bibitem{fsw96} R. C. Flagg, Ph. S"{u}nderhauf and K. R. Wagner,
{it A logical approach to quantitative domain theory}, Topology Atlas Preprint,
{bf 23} (1996), 10--29.

bibitem{gsw99} B. Ganter, G. Stumme and R. Wille, {it Formal concept analysis:
Mathematical Foundations}, Springer, Berlin, 1999.

bibitem{jag67} J. A. Goguen, {it $L$-fuzzy sets}, J. Math. Anal. Appl.,
 {bf 18} (1967), 145--174.

bibitem{fuzzy10} R. Gonz'{a}lez-del-Campo, L. Garmendia and B. De Baets,
{it Transitive closure of $L$-fuzzy relations and interval-valued fuzzy relations},
Fuzzy Systems (FUZZ), 2010 IEEE International Conference on, July (2010), 1--8.

bibitem{rh00} R. Halav{s}, {it Congruences on posets}, Contributions to
General Algebra, {bf 12} (2000), 195--210.

bibitem{hd03} R. Halav{s} and D. Hort, {it A characterization of
1-,2-,3-,4-homomorphisms of ordered sets}, Czechoslovak Math. J.,
{bf 53}textbf{(128)} (2003), 213--221.

bibitem{hol85} U. H"{o}hle and N. Blanchard, {it Partial ordering in $L$-under
determinate sets}, Information Science, {bf 35} (1985), 133--144.

bibitem{pk05} P. K"{o}rtesi, {it Congruences and isotone maps on partially
ordered sets}, Mathematica Pannonica, {bf 16}textbf{(1)} (2005), 39--55.

bibitem{lzh07} H. Lai and D. Zhang, {it Complete and directed complete
$Omega$-categories}, Theor. Comput. Sci., {bf 388} (2007), 1--25.

bibitem{mar11} P. Martinek, {it Completely lattice $L$-ordered sets with
and without $L$-equality}, Fuzzy Sets and Systems, {bf 166} (2011), 44--55.

bibitem{szk08} K. P. Shum, P. Zhu and N. Kehayopulu, {it III-Homomorphisms and
III-congruences on posets}, Discrete Math., {bf 308} (2008), 5006--5013.

bibitem{et99} E. Turunen, {it Mathematics behind fuzzy logic},
Physica--Verlag, 1999.

bibitem{ven92} P. Venugopalan, {it Fuzzy ordered sets}, Fuzzy Sets and Systems,
{bf46} (1992), 221--226.

bibitem{wag94} K. R. Wagner, {it Solving recursive domain equations with
enriched categories}, Ph.D. Thesis, School of Computer Science, Carnegie
Mellon University, Technical Report CMU-CS-94-159, July 1994.

bibitem{wag97} K. R. Wagner, {it Liminf convergence in $Omega$-categories},
Theor. Comput. Sci., {bf 184} (1997), 61--104.

bibitem{wd39} M. Ward and R. P. Dilworth, {it Residuated lattices},
T. Am. Math. Soc., {bf 45} (1939), 335--354.

bibitem{xzhf09} W. Xie, Q. Zhang and L. Fan, {it The Dedekind-MacNeille
completions for fuzzy posets}, Fuzzy Sets and Systems, {bf 160} (2009), 2292--2316.

bibitem{xs05} X. Xie and X. Shi, {it Order-congruences on S-posets}, Commun.
Korean Math. Soc., {bf 20}textbf{(1)} (2005), 1--14.

bibitem{yao10} W. Yao, {it Quantitative domains via fuzzy sets: part I:
continuity of fuzzy directed complete posets}, Fuzzy Sets and Systems,
{bf 161} (2010), 973--987.

bibitem{zad71} L. A. Zadeh, {it Similarity relations and fuzzy orderings},
Information Science, {bf 3} (1971), 177--200.

bibitem{zf05} Q. Zhang and L. Fan, {it Continuity in quantitative domains},
Fuzzy Sets and Systems, {bf 154} (2005), 118--131.

bibitem{zxf09} Q. Zhang, W. Xie and L. Fan, {it Fuzzy complete lattices},
Fuzzy Sets and Systems, {bf 160} (2009), 2275--2291.