Fuzzy posets can be regarded as a generalization of classical posets. Many concepts and results in posets can be generalized to fuzzy posets. In order to further improve the fuzzy poset theory, in this paper, we shall introduce the concept of prime (distributive) fuzzy posets which can be seen as the correspondence of prime (distributive) posets in fuzzy setting. First, we consider the extensions of fuzzy ideals in a fuzzy poset. Then by the extensions of fuzzy ideals, we give a characterization for prime (distributive) fuzzy posets, that is, a fuzzy poset X is prime (distributive) if and only if all extensions of normal fuzzy ideals of X are Frink (strong) fuzzy ideals. This paper generalizes the work of Halas in posets into fuzzy posets.
R.Bělohlávek, Lattices of fixed points of fuzzy Galois connection, Mathematical Logic Quarterly, 47 (2001), 111-116.
2.
R.Bělohlávek, Fuzzy closure operators, Journal of Mathematical Analysis and Applications, 262 (2001), 473–489.
3.
R.Bělohlávek, Fuzzy Relational Systems: Foundations and Principles, New York: Kluwer Acdemic/Plenum Publisher, (2002).
4.
R.Bělohlávek, Some properties of residuated lattices, Czechoslovak Mathematical Journal, 53 (2003), 161–171.
5.
R.Bělohlávek, Concept lattices and order in fuzzy logic, Annals of Pure and Applied Logic, 128 (2004), 277–298.
6.
U.Bodenhofer, A similarity-based generalization of fuzzy orderings preserving the classical axioms, International Journal of Uncertainty, Fuzziness Knowledge-Based Systems, 8 (2000), 593–610.
7.
U.Bodenhofer, Representations and constructions of similarity-based fuzzy orderings, Fuzzy Sets and Systems, 137 (2003), 113–136.
8.
M.Demirci, Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, part I: fuzzy functions and their applications, International Journal of General Systems, 32 (2003), 123–155.
9.
M.Demirci, A theory of vague lattices based on many-valued equivalence relations – I: general representation results, Fuzzy Sets and Systems, 151 (2005), 437–472.
10.
M.Demirci, Z.Eken, An introduce to vague complemented ordered sets, Information Sciences, 177 (2007), 150–160.
11.
L.Fan, A new approach to quantitative domain theory, Electronic Notes in Theoretical Computer Science, 45 (2001), 77–87.
J.A.Goguen, L-fuzzy sets, Journal of Mathematical Analysis and Applications, 18 (1967), 145–174.
14.
G.Grätzer, General Lattice Theory, Academic Press, New York, 1978.
15.
R.Halaš, On extensions of ideals in posets, Discrete Mathematics, 308 (2008), 4972–4977.
16.
U.Höhle, N.Blanchard, Partial ordering in L-underdeterminate sets, Information Sciences, 35 (1985), 133–144.
17.
X.Ma, J.Zhan, M.I.Ali, N.Mehmood, A survey of decision marking methods based on two classes of hybird soft set models, Artificial Intelligence Review, 49 (2018), 511–539.
18.
W.X.Xie, Q.Y.Zhang, L.Fan, The Dedekind-MacNeille completions for fuzzy posets, Fuzzy Sets and Systems, 160 (2009), 2292–2316.
W.Yao, Quantitative domain via fuzzy sets: Part I: Continuity of fuzzy directed complete posets, Fuzzy Sets and Systems, 161 (2010), 973-987.
21.
J.M.Zhan, W.H.Xu, Two types of coverings based multigranulation rough fuzzy sets and applications to decision making, Artificial Intelligence Review, 2018, https://doi.org/10.1007/s10462-018-9649-8.
22.
J.M.Zhan, Q.M.Wang, Certain types of soft coverings based rough sets with applications, International Journal of Machine Learning and Cybernetics, 2018, https://doi.org/10.1007/s13042-018-0785-x.
23.
L.Zhang, J.M.Zhan, J. C. R.Alcantud, Novel classes of fuzzy soft, A-coverings-based fuzzy rough sets with applications to multi-criteria fuzzy group decision making, Soft Computing, 2018, https://doi.org/10.1007/s00500-018-3470-9.
24.
L.Zhang, J.M.Zhan, Fuzzy soft, A-covering based fuzzy rough sets and corresponding decision-making applications, International Journal of Machine Learning and Cybernetics, 2018, https://doi.org/10.1007/s13042-018-0828-3.
25.
Q.Y.Zhang, L.Fan, Continuity in quantitative domains, Fuzzy Sets and Systems, 154 (2005), 118–131.
26.
Q.Y.Zhang, W.X.Xie, Section-retraction-pairs between fuzzy domains, Fuzzy Sets and Systems, 158 (2007), 99-114.
27.
L.A.Zadeh, Similarity relations and fuzzy orderings, Information Sciences, 3 (1971), 177–200.
