Geodesic
A binary operation-based representation of a lattice
Kybernetika, Tome 55 (2019) no. 2, pp. 252-272
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper, we study and characterize some properties of a given binary operation on a lattice. More specifically, we show necessary and sufficient conditions under which a binary operation on a lattice coincides with its meet (resp. its join) operation. Importantly, we construct two new posets based on a given binary operation on a lattice and investigate some cases that these two posets have a lattice structure. Moreover, we provide some representations of a given lattice based on these new constructed lattices.
In this paper, we study and characterize some properties of a given binary operation on a lattice. More specifically, we show necessary and sufficient conditions under which a binary operation on a lattice coincides with its meet (resp. its join) operation. Importantly, we construct two new posets based on a given binary operation on a lattice and investigate some cases that these two posets have a lattice structure. Moreover, we provide some representations of a given lattice based on these new constructed lattices.
DOI : 10.14736/kyb-2019-2-0252
Classification : 06B05, 06B15
Keywords: lattice; binary operation; neutral element; lattice representation 
@article{10_14736_kyb_2019_2_0252,
     author = {Yettou, Mourad and Amroune, Abdelaziz and Zedam, Lemnaouar},
     title = {A binary operation-based representation of a lattice},
     journal = {Kybernetika},
     pages = {252--272},
     year = {2019},
     volume = {55},
     number = {2},
     doi = {10.14736/kyb-2019-2-0252},
     mrnumber = {4014586},
     zbl = {07144937},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14736/kyb-2019-2-0252/}
}
TY  - JOUR
AU  - Yettou, Mourad
AU  - Amroune, Abdelaziz
AU  - Zedam, Lemnaouar
TI  - A binary operation-based representation of a lattice
JO  - Kybernetika
PY  - 2019
SP  - 252
EP  - 272
VL  - 55
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.14736/kyb-2019-2-0252/
DO  - 10.14736/kyb-2019-2-0252
LA  - en
ID  - 10_14736_kyb_2019_2_0252
ER  - 
%0 Journal Article
%A Yettou, Mourad
%A Amroune, Abdelaziz
%A Zedam, Lemnaouar
%T A binary operation-based representation of a lattice
%J Kybernetika
%D 2019
%P 252-272
%V 55
%N 2
%U http://geodesic.mathdoc.fr/articles/10.14736/kyb-2019-2-0252/
%R 10.14736/kyb-2019-2-0252
%G en
%F 10_14736_kyb_2019_2_0252
Yettou, Mourad; Amroune, Abdelaziz; Zedam, Lemnaouar. A binary operation-based representation of a lattice. Kybernetika, Tome 55 (2019) no. 2, pp. 252-272. doi: 10.14736/kyb-2019-2-0252

[1] Ashraf, M., Ali, S., Haetinger, C.: On derivations in rings and their applications. Aligarh Bull. Math. 25 (2006), 79-107. | MR

[2] Bede, B.: Mathematics of Fuzzy Sets and Fuzzy Logic. Springer, Berlin 2013. | MR | Zbl

[3] Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Springer, Heidelberg 2007.

[4] Birkhoff, G.: Lattice Theory. Third edition. Amer. Math. Soc., Providence 1967. | MR

[5] Blyth, T. S.: Set theory and abstract algebra. Longman, London, New York 1975. | MR

[6] Cooman, G. D., Kerre, E. E.: Order norms on bounded partially ordered sets. J. Fuzzy Math. 2 (1994), 281-310. | MR | Zbl

[7] Davey, B. A., Priestley, H. A.: Introduction to Lattices and Order. Second edition. Cambridge University Press, 2002. | DOI | MR

[8] Dummit, D. S., Foote, R. M.: Abstract Algebra. Third edition. Hoboken, Wiley 2004. | MR

[9] Ferrari, L.: On derivations of lattices. Pure Math. Appl. 12 (2001), 365-382. | MR

[10] Grätzer, G., Wehrung, F.: Lattice Theory: Special Topics and Applications. Volume 1. Springer International Publishing Switzerland, 2014. | DOI | MR

[11] Grätzer, G., Wehrung, F.: Lattice theory: Special Topics and Applications. Volume 2. Springer International Publishing Switzerland, 2016. | DOI | MR

[12] Halaš, R., Pócs, J.: On the clone of aggregation functions on bounded lattices. Inform. Sci. 329 (2016), 381-389. | DOI

[13] Jwaid, T., Baets, B. De, Kalická, J., Mesiar, R.: Conic aggregation functions. Fuzzy Sets Systems 167 (2011), 3-20. | DOI | MR

[14] Karaçal, F., glu, M. N. Kesicio\v: A t-partial order obtained from t-norms. Kybernetika 47 (2011), 300-314. | MR

[15] Karaçal, F., Mesiar, R.: Aggregation functions on bounded lattices. Int. J. General Systems 46 (2017), 37-51. | DOI | MR

[16] Kolman, B., Busby, R. C., Ross, S. C.: Discrete Mathematical Structures. Fourth edition. Prentice-Hall, Inc., 2003.

[17] Komorníková, M., Mesiar, R.: Aggregation functions on bounded partially ordered sets and theirs classification. Fuzzy Sets Systems 175 (2011), 48-56. | DOI | MR

[18] Lidl, R., Pilz, G.: Applied Abstract Algebra. Second edition. Springer-Verlag, New York, Berlin, Heidelberg 1998. | DOI | MR

[19] Lipschutz, S.: Discrete Mathematics. Third edition. McGraw-Hill, 2007. | DOI

[20] Martínez, R., Massó, J., Neme, A., Oviedo, J.: On the lattice structure of the set of stable matchings for a many to one model. Optimization 50 (2001), 439-457. | DOI | MR

[21] Medina, J.: Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices. Fuzzy Sets and Systems 202 (2012), 75-88. | DOI | MR

[22] Mesiar, R., Komorníková, M.: Aggregation functions on bounded posets. 35 Years of Fuzzy Set Theory, Springer, Berlin, Heidelberg 261 (2010), pp. 3-17. | DOI | MR

[23] Ponasse, D., Carrega, J. C.: Algèbre et tobologie boléennes. Masson, Paris 1979. | MR

[24] Risma, E. P.: Binary operations and lattice structure for a model of matching with contracts. Math. Soc. Sci. 73 (2015), 6-12. | DOI | MR

[25] Roman, S.: Lattices and Ordered Sets. Springer Science and Business Media, New York 2008. | DOI | MR

[26] Rosenfeld, A.: An Introduction to Algebraic Structures. Holden-Day, San Francisco 1968. | MR

[27] Schröder, B. S.: Ordered Sets. Birkhauser, Boston 2003. | DOI | MR

[28] Szász, G.: Translationen der verbände. Acta Fac. Rer. Nat. Univ. Comenianae 5 (1961), 449-453. | DOI | MR

[29] Szász, G.: Derivations of lattices. Acta Sci. Math. 37 (1975), 149-154. | DOI | MR

[30] Xin, X. L., Li, T. Y., Lu, J. H.: On derivations of lattices. Inform. Sci. 178 (2008), 307-316. | DOI | MR

Cité par Sources :