In this paper, we further investigate the constructions of implications and left (right) semi-uninorms on a complete lattice. We firstly give out the formulas for calculating the upper and lower approximation conjunctive left (right) semi-uninorms of a binary operation. Then, we derive the formulas for calculating the upper and lower approximation NP implications of a binary operation. Finally, we show that the right (left) residuum of the upper approximation conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorm of a right (left) infinitely ∨-distributive binary operation is the lower approximation right infinitely ∧-distributive NP implication of the right (left) residuum of the binary operation.
AlsinaC., FrankM.J., and SchweizerB., Associative Functions: Triangular Norms and Copulas, World Scientific Publishing Company, Singapore, 2006.
2.
BaczyńskiM., and JayaramB., Fuzzy Implication, Studies in Fuzziness and Soft Computing, Vol. 231, Springer, Berlin, 2008.
3.
BaczyńskiM., and JayaramB., QL-implications: Some properties and intersections, Fuzzy Sets and Systems161 (2010), 158–188.
4.
BirkhoffG., Lattice Theory, American Mathematical Society Colloquium Publishers, Providence, 1967.
5.
BurrisS., and SankappanavarH.P., A Course in Universal Algebra, World Publishing Corporation, Beijing, 1981.
6.
BustinceH., BurilloP., and SoriaF., Automorphisms, negations and implication operators, Fuzzy Sets and Systems134 (2003), 209–229.
7.
De BaetsB., and FodorJ., Residual operators of uninorms, Soft Computing3 (1999), 89–100.
8.
De CoomanG., and KerreE.E., Order norms on bounded partially ordered sets, J Fuzzy Math2 (1994), 281–310.
9.
DuranteF., KlementE.P., MesiarR., and SempiC., Conjunctors and their residual implicators: Characterizations and construction methods, Mediterranean Journal of Mathematics4 (2007), 343–356.
10.
FodorJ.C., Strict preference relations based on weak t-norms, Fuzzy Sets and Systems43 (1991), 327–336.
11.
FodorJ.C., and KeresztfalviT., Nonstandard conjunctions and implications in fuzzy logic, International Journal of Approximate Reasoning12 (1995), 69–84.
12.
FodorJ.C., and RoubensM., Fuzzy Preference Modelling and Multicriteria Decision Support, Theory and Decision Library, Series D: System Theory, Knowledge Engineering and Problem Solving, Kluwer Academic Publishers, Dordrecht, 1994.
13.
FodorJ.C., YagerR.R., and RybalovA., Structure of uninorms, Internat J Uncertainty, Fuzziness and Knowledge-Based Systems5 (1997), 411–427.
14.
GabbayD., and MetcalfeG., Fuzzy logics based on [0; 1)-continuous uninorms, Arch Math Logic46 (2007), 425–449.
15.
HaoX.Y., NiuM.X., and WangZ.D., The relations between implications and left (right) semi-uninorms on a complete lattice, Internat J Uncertainty, Fuzziness and Knowledge-Based Systems23 (2015), 245–261.
16.
JeneiS., and MontagnaF., A general method for constructing leftcontinuous t-norms, Fuzzy Sets and Systems136 (2003), 263–282.
17.
KlementE.P., MesiarR., and PapE., Triangular Norms, Trends in Logic, Studia Logica Library, Vol. 8, Kluwer Academic Publishers, Dodrecht,
2000.
18.
LiZ.B., and LiuH.W., Generalization of (U, N)-implications derived from commutative semiuninorms and pseudo-uninorms, Journal of Intelligent and Fuzzy Systems29 (2015), 2177–2184.
19.
LiuH.W., Semi-uninorm and implications on a complete lattice, Fuzzy Sets and Systems191 (2012), 72–82
.
20.
MasM., MassanetS., Ruiz-AguileraD., and TorrensJ., A survey on the existing classes of uninorms, Journal of Intelligent and Fuzzy Systems29 (2015), 1021–1037.
21.
MasM., MonserratM., and TorrensJ., On left and right uninorms, Internat J Uncertainty, Fuzziness and Knowledge-Based Systems9 (2001), 491–507.
22.
MasM., MonserratM., and TorrensJ., On left and right uninorms on a finite chain, Fuzzy Sets and Systems146 (2004), 3–17.
23.
MasM., MonserratM., and TorrensJ., Two types of implications derived from uninorms, Fuzzy Sets and Systems158 (2007), 2612–2626.
24.
MusilekP., GuanlaoR., and BarreiroG., Genetic programming of fuzzy aggregation operations, Journal of Intelligent and Fuzzy Systems16 (2005), 107–118.
25.
NiuM.X., HaoX.Y., and WangZ.D., Relations among implications, coimplications and left (right) semi-uninorms, Journal of Intelligent and Fuzzy Systems29 (2015), 927–938
.
26.
OuyangY., On fuzzy implications determined by aggregation operators, Information Sciences193 (2012), 153–162.
27.
ShiY., Van GasseB., RuanD., and KerreE.E., On dependencies and independencies of fuzzy implication axioms, Fuzzy Sets and Systems161 (2010), 1388–1405.
28.
SuY., and WangZ.D., Pseudo-uninorms and coimplications on a complete lattice, Fuzzy Sets and Systems224 (2013), 53–62.
29.
SuY., and WangZ.D., Constructing implications and coimplications on a complete lattice, Fuzzy Sets and Systems247 (2014), 68–80.
30.
SuY., WangZ.D., and TangK.M., Left and right semi-uninorms on a complete lattice, Kybernetika49 (2013), 948–961.
31.
Suárez GarcíaF., and Gil ÁlvarezP., Two families of fuzzy intergrals, Fuzzy Sets and Systems18 (1986), 67–81.
32.
TakácsM., Uninorm-based models for FLC systems, Journal of Intelligent and Fuzzy Systems19 (2008), 65–73.
33.
TsadirasA.K., and MargaritisK.G., The MYCIN certainty factor handling function as uninorm operator and its use as a threshold function in artificial neurons, Fuzzy Sets and Systems93 (1998), 263–274.
34.
WangZ.D., Generating pseudo-t-norms and implication operators, Fuzzy Sets and Systems157 (2006), 398–410.
35.
WangZ.D., Left (right) semi-uninorms and coimplications on a complete lattice, Fuzzy Sets and Systems287 (2016), 227–239.
36.
WangZ.D., and FangJ.X., Residual operators of left and right uninorms on a complete lattice, Fuzzy Sets and Systems160 (2009), 22–31.
37.
WangZ.D., and FangJ.X., Residual coimplicators of left and right uninorms on a complete lattice, Fuzzy Sets and Systems160 (2009), 2086–2096.
38.
WangZ.D., and YuY.D., Pseudo-t-norms and implication operators on a complete Brouwerian lattice, Fuzzy Sets and Systems132 (2002), 113–124.
39.
YagerR.R., Uninorms in fuzzy system modeling, Fuzzy Sets and Systems122 (2001), 167–175.
40.
YagerR.R., and RybalovA., Uninorm aggregation operators, Fuzzy Sets and Systems80 (1996), 111–120.
